Answer to Question #4505 in Algebra for ramu

Question

How to find reminder of (12^107)/37. please let me know

Answer 1

12^107=12^(3*36)* 12^-1
12^36=1 (mod 37) => 12^(3*36)=1 (mod 37)
12^107=12^-1(mod 37)
12^-1=34(mod 37)
We use here Euler’s theorem: a^f(m)=1 (mod m) <=> gcd(a,m)=1 in second row
answer: 34