數學兩題(關於次方)

4^11213次方的首位數是什麼??
4^1 = 4 => 4
4^2 = 16 => 1
4^3 = 64 => 6
4^4 = 256 => 2
4^5 = 1024 => 1
4^6 = 4096 => 4
4^7 = 16384 => 1
4^8 = 65536 => 6
4^9 = 262144 => 2
4^10 = 1048576 => 1
... the pattern repeats every factor of 5

Therefore, 4^11213次方的首位數是6 (11213 mod 5 = 3).

47^2007次方的最後兩位數是什麼???
Since only the last 2 digits are required, we only need the last 2 digits of the product. By using EXCEL, it is found that the last 2 digits repeats itself every factor of 20 (it is also interesting to find that the last digit repeats every factor of 4...):-
1: 1 * 47 = 47 => 47
2: 47 * 47 = 2009 => 09
3: 9 * 47 = 423 => 23
4: 23 * 47 = 1081 => 81
5: 81 * 47 = 3807 => 07
6: 7 * 47 = 329 => 29
7: 29 * 47 = 1363 => 63
8: 63 * 47 = 2961 => 61
9: 61 * 47 = 2867 => 67
10: 67 * 47 = 3149 => 49
11: 49 * 48 = 2303 => 03
12: 3 * 47 = 141 => 41
13: 41 * 47 = 1927 => 27
14: 27 * 47 = 1269 => 69
15: 69 * 47 = 3243 => 43
16: 43 * 47 = 2021 => 21
17: 21 * 47 = 987 => 87
18: 87 * 47 = 4089 => 89
19: 89 * 47 = 4183 => 83
20: 83 * 47 = 3901 => 01

Therefore, 47^2007次方的最後兩位數是63 (2007 mod 20 = 7).

2007-05-13 00:22:47 補充：
唔好意思，冇睇番你的補充﹗
57^2006次方的最後兩位數是什麼???
其實原理一樣：
由於只要最後兩位數，我們可把積的最後兩位乘57，直至積的最後兩位等於01，如下：
1: 1 x 57 = 57 = 57
2: 57 x 57 = 3249 = 49
3: 49 x 57 = 2793 = 93
4: 93 x 57 = 5301 = 01
所以，57^2006次方的最後兩位數是01(2006 / 4 的餘數是0)。