Question

Remainder when 66666....45 times when divided by 37

Answer 1

By Fermat’s ‘‘‘‘little”” theorem, if pp is prime and a∈Za∈Z is such that p∤ap∤a, then ap−1≡1(modp)ap−1≡1(modp). Applying this to p=79p=79and a=37a=37,

37157=37⋅(3778)2≡37(mod79)37157=37⋅(3778)2≡37(mod79).

The remainder is 3737. ■