Track / Stage 5 / 199 of 400 #799 of 1964

Problem 799

AIME late

Number theory Difficulty 5.5 Prove it

[ Fermat's Little Theorem ]

Prove that the number $30^{239} + 239^{30}$ is composite.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

This number is divisible by 31.

## Solution

$30^{239}+239^{30} \equiv(-1)^{239}+1=0(\bmod 31)$ .

Let $p$ be a prime number. Prove that $(a+b)^{p} \equiv a^{p}+b^{p}(\bmod p)$ for any integers $a$ and $b$ .

## Solution

$(a+b)^{p} \equiv a+b \equiv a^{p}+b^{p}(\bmod p)$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.