Question #4505

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

Expert's answer

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

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

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