Question

find the remainder when 2 power 89 is divided by 89

Answer 1

Answer: 2

Fermat’s little theorem

If p is a prime number and a is a natural number, then

a^p=a(mod p)…………i

Rem(a^p/p)=a

This theorem can also be stated as: If p is a prime number and a is co prime to p, then

a^ p -1 ≡ 1 (mod p)………….ii

Rem(a^p-1/p)=1

so from (ii)

2^88=1(mod 89)

from (i)

2^1=2(mod)89

2^89 = 2(mod) 89

therefore remainder = 2