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## Primes and divisibility

Prove the following without using Fermat's little theorem or Euler's totient theorem. Here $a\ |\ b$ means $a$ divides $b$.

1. Prove that for every prime number $p$ and every pair of integers $a$ and $b$,

$p\ |\ ((a+b)^p -a^p -b^p)$

2. Prove that for every integer $a$ and every prime number $p$

$p\ |\ (a^p - a).$

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