# Remainder of $7^{220}$ when divided by $8$

How can I find out what is the remainder when I divide $7^{220}$ by $8$ using modular arithmetic and without using any theorems such as Fermat's Little Theorem or Chinese Remainder Theorem?

• Any odd square is congruent to $1$ modulo $8$. – Quang Hoang Nov 13 '15 at 3:37

Hint. $7\equiv -1 \pmod 8$, and $220$ is even ...