What are the final two digits of 71997?
70 = 1 (consider that as 01)
71 = 7 (consider that as 07)
72 = 49
73 = 343
74 = 2401
75 = 16807
76 = 117649
77 = 823543
78 = 5764801
...
So we see that the last two digits of powers of 7 are
01, 07, 49, 43, 01, 07, 49, 43,
ad the last two digits continue to repeat in groups of 4 forever.
So to find the last two digits of any positive integral power of
7, we can just divide the power by 4 and take only the remainder.
If the remainder is 0, the last two digits are 01,
If the remainder is 1, the last two digits are 07,
If the remainder is 2, the last two digits are 49,
If the remainder is 3, the last two digits are 43,
To find 71997, we divide 1997 by 4
499
4)1997
16
39
36
37
36
1
The remainder is 1, so the last two digits of 71997 are 07.
Edwin
