Question

Prove that the number A= [tex]20^{8^{2014} } }+113 is composite

Answer 1

Cheese proof:

We can just prove that it is divisible by 3, which means that it is composite. We can use modular exponentiation, where if
`a \equiv b \pmod{n}` then
`a^x \equiv b^x \pmod{n}`. In this case,
`{-1}^{8^(2014)} \equiv 20^{8^(2014)} \pmod{3}`. This is much easier to calculate! Since
`8^(2014)` is even,
`-1^{8^(2014)}=1`, meaning that now we only need to prove that
`0\equiv(1+113) \pmod{3}`, which is obviously true.