# Fermat's little theorem

## Proof

If $$a$$ is an integer, $$p$$ is a prime number, and $$a$$ is not divisible by $$p,$$ then $$a^{p - 1} \equiv 1 \pmod{p}$$
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If $$a$$ is an integer, $$p$$ is a prime number, and $$a$$ is not divisible by $$p,$$ then $$a^p \equiv a \pmod{p}$$

## Using the theorem

Find the remainder: $$2^{50} \div 17$$
$$4$$
Find the remainder: $$4^{532} \div 11$$
$$5$$
Find the remainder: $$7^{2001} \div 5$$
Find the remainder: $$5^{3571} \div 11$$
Find the remainder: $$4^{100{,}000} \div 19$$