# Tagged: congruence

## Problem 344

Let $a, b$ be relatively prime integers and let $p$ be a prime number.
Suppose that we have
$a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ for some positive integer $n$.

Then prove that $2^{n+1}$ divides $p-1$.

## Problem 219

Use Lagrange’s Theorem in the multiplicative group $(\Zmod{p})^{\times}$ to prove Fermat’s Little Theorem: if $p$ is a prime number then $a^p \equiv a \pmod p$ for all $a \in \Z$.