Thursday, 18 July 2019

Modular exponentiation problem




107(mod77)



I tried repeated squaring, which worked but took many computations. I also tried Fermat's little theorem, but since 7<77 I didn't know how to use it.



Any simpler way to do this?


Answer



Not much effort saved, but work separately mod 11 (easy, since 101(mod11)) and mod 7 (easy since by Fermat 10710(mod7)). Then stitch the answers together using the Chinese Remainder Theorem. In this case that can be done by inspection: the answer is 10.


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