Question

What are the final two digits of 7^177

Answer 1

Phi(100) = 40, so the order of 7 (i.e. the smallest power to which you have to raise 7 to get 1) is a divisor of 40, and you easily find that it is 4.

Computing Mod 100 in the following:

7^4 = 2401 = 1

176 is a multiple of 4, so we have:

7^177 = 7

Answer 2

Hint: take a look at the last two digits for the first few powers of 7

7^1 = 7 or 07
7^2 = 49
7^3 = 343
7^4 = 2401
7^5 = 16807
7^6 = ......49
7^7 = ......43
7^8 = ......01
7^9 = ......07
7^10 = .... 49

I am sure you can do something with this pattern.