Congruence to a power

Question

I know a^m = 1 mod p How do I know that a^(mp)= 1 mod p^2?

Can someone please explain to me some rules of congruence regarding powers? Thank you.

Answer 1

$$a^m\equiv1\pmod p\implies a^m=1+kp$$ for some integer $k$

Now $a^{mp}=(a^m)^p=(1+kp)^p=1+kp^2+\text{ terms containing }p^2\equiv1\pmod{p^2}$