Prove: $2005|\underbrace{5 5\cdots 5}_{800\text{ digits}}$

Question

$2005=5\cdot 401$

Divisibility by $5$ follows from the last digit of $\underbrace{5 5\cdots 5}_{800\text{ digits}}$.

How toprove divisibility by $401$?

Answer 1

Your number is $5\left(\dfrac{10^{800} - 1}{9} \right)$

$401$ is a prime number $\implies 10^{400} \equiv 1 \pmod{401} \implies 10^{800} \equiv 1 \pmod{40}$

$\implies 5\left( \dfrac{10^{800} - 1}{9} \right) \equiv 0 \pmod{401}$