Show that $a^{2001} \equiv a \pmod {1000}$

Question

Suppose $a$ is a positive integer which is coprime to 10. Show that $$a^{2001} \equiv a \pmod {1000}$$

I know it has something to do with the Fermat-Euler theorem.

$\phi(1000) = 400$ and $a^{400}\equiv 1 \pmod {1000}$

However, I do not know how to proceed from here to show the congruence

Answer 1

$a^{2000}≡a^{5\cdot400}=\left(a^{400}\right)^5≡1 (\mod 1000)$ therefore $^{2001}≡^{2000}≡[1]≡$

Answer 2

As $1000=2^35^3$

Using Carmichael Function

$\lambda(5^32^3)=100,5^3$ will divide $a^n-1$ if $100|n$ and $(a,10)=1$