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.