Posted by **Anonymous** on Friday, August 3, 2012 at 10:01pm.

What are the final two digits of 7^177

- math -
**Count Iblis**, Friday, August 3, 2012 at 11:35pm
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

- math -
**Reiny**, Friday, August 3, 2012 at 11:37pm
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.

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