Như bạn có thể thấy, sức mạnh tăng lên nhanh chóng. Vì vậy, thật hữu ích khi nhận thấy rằng chúng ta chỉ cần xem xét chữ số hàng đơn vị khi đánh giá các lũy thừa lớn. Ví dụ: chữ số hàng đơn vị của 132 bằng chữ số hàng đơn vị của 32, chữ số hàng đơn vị của 135 bằng chữ số hàng đơn vị của 35, v.v.
Tiếp tục, ta được
131 có chữ số hàng đơn vị là 3
132 có chữ số hàng đơn vị là 9
133 có chữ số hàng đơn vị là 7
134 có chữ số hàng đơn vị là 1
135 có chữ số hàng đơn vị là 3
136 có chữ số hàng đơn vị là 9
137 có chữ số hàng đơn vị là 7
138 có chữ số hàng đơn vị là 1
Lưu ý rằng một mô hình đẹp xuất hiện. Chúng tôi nhận được. 3-9-7-1-3-9-7-1-3-9-7-1-…
Như bạn có thể thấy, mô hình lặp lại sau mỗi 4 lũy thừa. Tôi muốn nói rằng "chu kỳ" bằng 4
Bây giờ chúng ta biết chu kỳ là 4, chúng ta có thể thực hiện một quan sát rất quan trọng
Khi n là bội số của 4 thì chữ số hàng đơn vị của 13n là 1
Đó là,
134 có chữ số hàng đơn vị là 1
138 có chữ số hàng đơn vị là 1
1312 có chữ số hàng đơn vị là 1
1316 có chữ số hàng đơn vị là 1
. . vân vân
Lúc này ta tìm được chữ số hàng đơn vị của 1335
Vì 32 là bội của 4 nên 1332 phải có chữ số hàng đơn vị là 1. Từ đây, chúng ta sẽ tiếp tục mô hình
Hệ thống số là một phương pháp biểu diễn các số trên dòng số với sự trợ giúp của một tập hợp các ký hiệu và quy tắc. Các ký hiệu này nằm trong khoảng từ 0-9 và được gọi là chữ số. Hệ thống Số được sử dụng để thực hiện các phép tính toán học, từ các phép tính khoa học tuyệt vời đến các phép tính như đếm số lượng Đồ chơi cho Trẻ em hoặc số lượng sôcôla còn lại trong hộp. Hệ thống số bao gồm nhiều loại dựa trên giá trị cơ bản cho các chữ số của chúng
Chu kỳ của các số. Chu kỳ của bất kỳ số nào chủ yếu tập trung vào chữ số hàng đơn vị của nó. Mỗi chữ số hàng đơn vị có kiểu lặp lại riêng khi nâng lên bất kỳ lũy thừa nào. Khái niệm này được sử dụng rất nhiều trong khi giải các bài toán năng khiếu. Khái niệm về chu kỳ của các số có thể được học bằng cách tìm ra các chữ số hàng đơn vị của tất cả các số có một chữ số từ 0 đến 9 khi được nâng lên các lũy thừa nhất định. Những con số này có thể được phân loại thành ba loại được liệt kê như sau
1. Các chữ số 0, 1, 5 và 6. Ở đây, khi mỗi chữ số này được nâng lên một lũy thừa bất kỳ, thì chữ số hàng đơn vị của đáp án cuối cùng chính là số đó
ví dụ
1. 5 ^ 2 = 25: Unit digit is 5, the number itself. 2. 1 ^ 6 = 1: Unit digit is 1, the number itself. 3. 0 ^ 4 = 0: Unit digit is 0, the number itself. 4. 6 ^ 3 = 216: Unit digit is 6, the number itself.
Dưới đây là một số câu hỏi dựa trên khái niệm trên
Câu hỏi 1. Tìm chữ số hàng đơn vị của 416345
Trả lời. Chỉ cần tìm 6345 sẽ cho 6 là chữ số hàng đơn vị, do đó chữ số hàng đơn vị của 416345 là 6
Câu hỏi 2. Tìm chữ số hàng đơn vị của 23534566
Trả lời. Tìm 534566 biết chữ số hàng đơn vị là 5 nên chữ số hàng đơn vị của 23534566 là 5
2. Chữ số 4 và 9. Cả hai chữ số 4 và 9 này đều có tích hai chữ số khác nhau là chữ số hàng đơn vị
ví dụ
1. 4 ^ 2 = 16: Unit digit is 6. 2. 4 ^ 3 = 64: Unit digit is 4. 3. 4 ^ 4 = 256: Unit digit is 6. 4. 4 ^ 5 = 1024: Unit digit is 4. 5. 9 ^ 2 = 81: Unit digit is 1. 6. 9 ^ 3 = 729: Unit digit is 9.
Có thể nhận thấy các chữ số hàng đơn vị 2, 4, 8, 6 lặp lại sau khoảng thời gian bốn số. Tương tự,
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 93
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 94
64
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 96
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 98
61
30
31
32
33
64
35
36
37_______10_______8
37_______10_______0
2 61
32
Java
2 63
2 64
2 65
2 66
2 67
2 68
61
// Efficient C++ program to
061
// Efficient C++ program to
2 64
65
64
67
64
69
61
60
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
62
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
64
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
64
// find unit digit of x^y.
7// find unit digit of x^y.
8// find unit digit of x^y.
961
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
69
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 91
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 93
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 94
64
#include
9using
0using
1using
2_______722_______3using
4// find unit digit of x^y.
9
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
30
31
61
32
61
61
33
61
606
// Efficient C++ program to
2 608
609
610
611
61
60
61
615
616
617
618
619
61
32
32
623
Python3
624
625
626
627
628
629
61
631
61_______2_______33
61
635
636
// find unit digit of x^y.
861
61
640
61
642
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 93
645_______2_______46
647
648
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
635_______2_______36
652_______2_______53
654
655
using
461
61
30
635
61
662
663
664
616
617
618
668
669
C#
2 63
2 64
using
673
2 67
675
60
61
// Efficient C++ program to
061
// Efficient C++ program to
2 64
65
64
67
64
69
61
60
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
62
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
64
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
64
67
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
69
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 91
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 93
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 94
64
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 96
using
2______3_______8
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
30
31
61
32
61
61
33
61
606
// Efficient C++ program to
2 608
621
61
60
61
625
61
32
32
629
PHP
630
631
// find unit digit of x^y.
62
63
635
65_______2_______37
617
639
640
60
61
62
61
64
61_______2_______47
648
61
69
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 91
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 93
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 94
656
657
656
659
639
// find unit digit of x^y.
9656
663
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
647
666
647
653
637
670
61
30
647
// find unit digit of x^y.
932
676
677
678
679
680
Javascript
681
682
// find unit digit of x^y.
62
63
635
687
60
61
61
62
61
64
61_______2_______95
61
69
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 91
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 93
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 902
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 98
61
30
31
32
676
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 910
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 911
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 912
Đầu ra
6
đầu ra.
6
Thời gian phức tạp. O[y], trong đó y là lũy thừa
Không gian phụ trợ. O[1], vì không cần thêm dung lượng
Tối ưu hóa hơn nữa. Chúng ta có thể tính toán sức mạnh mô-đun trong Log y
Cách 3 [Trực tiếp dựa trên tính chất tuần hoàn của chữ số cuối cùng]
Phương pháp này phụ thuộc vào chu kỳ với chữ số tận cùng của x là
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 9
Vì vậy, ở đây chúng tôi trực tiếp sửa đổi lũy thừa y bằng 4 vì đây là lũy thừa cuối cùng sau khi bắt đầu lặp lại tất cả các số này
sau đó, chúng ta chỉ cần cấp nguồn cho số x chữ số cuối cùng, sau đó chúng ta nhận được chữ số hàng đơn vị của số được sản xuất.
C++
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 913
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 914
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 915
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 916
using
60
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 920
64
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 922_______2_______4
67
64
69
60
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 929
61_______3_______31
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 932
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 934
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 936
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 937
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 939
61_______3_______41
61_______3_______43
61_______3_______45
61
30
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 948_______2_______4
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 950
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 951
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 952
32
64
35
60
61
64
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 959
61
61____3_______62
61____3_______64
61____3_______66
61____3_______68
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 969
61
30
2 61
32
Java
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 974
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 975
2 65
2 66
2 65
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 979
2 67
2 68
61
61_______3_______20
61
// Efficient C++ program to
2 64
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 922
64
67
64
69
61
60
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 929
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 998
using
4// find unit digit of x^y.
9using
2
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 934
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 936
306
using
0640
309_______10_______10
616
312
616
// find unit digit of x^y.
961
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 941
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 943
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 945
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
30
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 948
64
326
using
4328
61
32
61
61
61
606
// Efficient C++ program to
2 608
337
61
60
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
64
342
343
344
345
// find unit digit of x^y.
961
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 962
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
351
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 966
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
355
61
61
32
32
360
Python3
361
362
2 65
364
365
628
367
61
369
61
371
636
371____2_______55
using
4x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
61
378
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 936
381
636
using
0384
385
386
636
386
655
616
312
616
61
394
61
396
61
398
61
30
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 948_______2_______4
2 603
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 951
2 605
655
using
4640
2 609
371
636
343
2 613
636
345
61
2 617
2 618
2 619
663
2 621
2 622
C#
2 623
2 624
using
673
2 67
2 68
61
61_______3_______20
61
// Efficient C++ program to
2 64
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 922
64
67
64
69
61
60
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 929
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 931
using
2
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 934
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 936
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 937
309______3_______39
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 941
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 943
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 945
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
30
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 948
64
2 665
61
32
61
61
2 670
61
606
// Efficient C++ program to
2 608
2 675
61
60
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
64
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 959
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 962
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
351
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 966
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
2 689
61
61
32
32
629
PHP
630
2 696
2 624
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 920
635
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 922
637
617_______2_______39
640
60
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 929
61
637
636
637
// Efficient C++ program to
1261
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 934
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 936
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 94
639
// Efficient C++ program to
20x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
639
636
639
// Efficient C++ program to
25
61_______3_______41
61_______3_______43
61_______3_______45
61
30
// Efficient C++ program to
34637
617
639
// Efficient C++ program to
3832
2 670
637
// Efficient C++ program to
42639
// Efficient C++ program to
4461
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 962
351
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 966
677
// Efficient C++ program to
50637
617
639
619
679
680
Javascript
681
// Efficient C++ program to
58
2 624
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 920
635
// Efficient C++ program to
6260
61
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 929
61_______3_______31
61
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 934
61_______3_______36
// Efficient C++ program to
74x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 939
61_______3_______41
61_______3_______43
61_______3_______45
61
30
// Efficient C++ program to
8532
2 670
// Efficient C++ program to
88
// Efficient C++ program to
89
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 962
351
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 966
// Efficient C++ program to
94
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 911
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 912
Đầu ra
3
Thời gian phức tạp. O[log n]
Không gian phụ trợ. Ô[1]
Tiếp cận. Phương pháp khai triển nhị thức
Sau đây là các bước tìm chữ số hàng đơn vị của x lũy thừa y bằng phương pháp Khai triển nhị thức
1. Xử lý các trường hợp đặc biệt
Nếu y bằng 0, trả về 1 vì bất kỳ số nào được nâng lên lũy thừa 0 đều bằng 1
Nếu x là 0, trả về 0 vì bất kỳ số nào lũy thừa 0 đều là 1 và chữ số hàng đơn vị của 0 là 0
2. Tính số hạng thứ y trong khai triển của [x+10]^y sử dụng định lý nhị thức
Thuật ngữ thứ y trong khai triển được cho bởi. C[y, 0]x^y10^0 + C[y, 1]*x^[y-1]*10^1 + … + C[y, y]x^010^y
Ở đây, C[y, k] đại diện cho hệ số nhị thức, bằng với y. / [k. * [y-k]. ]
Ta chỉ cần tính số hạng cuối cùng trong khai triển này là C[y, y]x^010^y
3. Tìm chữ số hàng đơn vị của số hạng thứ y
Chữ số hàng đơn vị của số hạng thứ y bằng chữ số tận cùng của số hạng thứ y
Chúng ta có thể tìm chữ số tận cùng của số hạng thứ y bằng cách lấy phần còn lại của số hạng đó khi chia cho 10
4. Trả về chữ số hàng đơn vị tìm được ở bước 3 là kết quả
C++
// Efficient C++ program to
97
// Efficient C++ program to
98
using
60
61
64
// find unit digit of x^y.
0364
67
64
// find unit digit of x^y.
0761_______3_______36
// find unit digit of x^y.
10x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
30
// find unit digit of x^y.
1361
32
61
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 936
// find unit digit of x^y.
18x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
30
2 61
61
32
61
64
// find unit digit of x^y.
26x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 951_______718_______28
61
64
// find unit digit of x^y.
3161
30
// find unit digit of x^y.
3432
64
// find unit digit of x^y.
3761
// find unit digit of x^y.
39// find unit digit of x^y.
4061
// find unit digit of x^y.
42// find unit digit of x^y.
4361
30
2 61
32
// find unit digit of x^y.
48
Java
2 65
// find unit digit of x^y.
50
606
2 67
// find unit digit of x^y.
5361
606
// Efficient C++ program to
2 64
// find unit digit of x^y.
5864
67
64
// find unit digit of x^y.
07x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 936
// find unit digit of x^y.
65using
0// find unit digit of x^y.
67using
2_______10_______0 // find unit digit of x^y.
8// find unit digit of x^y.
9
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
32
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 936
// find unit digit of x^y.
76using
0// find unit digit of x^y.
67using
2_______10_______0 using
0// find unit digit of x^y.
9
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
32
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
64
// find unit digit of x^y.
8764_______718_______89
using
4// find unit digit of x^y.
91x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
64
// find unit digit of x^y.
94using
4// find unit digit of x^y.
9x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
30
// find unit digit of x^y.
9961
32
61
606
// Efficient C++ program to
2 608
#include
06x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
#include
08618
617
// find unit digit of x^y.
8619
// find unit digit of x^y.
40x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
#include
08616
617
618
619
// find unit digit of x^y.
4361
32
32
Python3
628
#include
2561
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 936
386
636_______2_______36
using
0384
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
30
// find unit digit of x^y.
861
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 936
371
636
636
using
0384
x | power 2 | power 3 | power 4 | Cyclicity 0 | ................................. | ... repeat with 0 1 | ................................. | ... repeat with 1 2 | 4 | 8 | 6 | ... repeat with 2 3 | 9 | 7 | 1 | ... repeat with 3 4 | 6 |...................... | ... repeat with 4 5 | ................................. | ... repeat with 5 6 | ................................. | ... repeat with 6 7 | 9 | 3 | 1 | ... repeat with 7 8 | 4 | 2 | 6 | ... repeat with 8 9 | 1 | ..................... | ... repeat with 97
30
using
061
#include
47636
#include
49312
using
4640
653
653
386