1. Modular

I have two questions involving mod m that I am struggling with and I was hoping someone could give me some tips on how to start and or complete them

1. if a = b(mod m) and a = b(mod n), prove a = b(mod mn), I know that m and n are relatively prime. So the greatest common divisor of m and n is 1

2. If x^2 = a^2 (mod p), prove x= a(mod p) or x = -a (mod p). Where p is a prime..

2. For $1)$, use the definition of congruence, and the fact that if $(m,n)=1$, $m\mid A$ and $n \mid A$ then $mn \mid A$.

For $2)$, notice that $x^2-a^2 \equiv 0 \mod p \Rightarrow (x-a)(x-b) \equiv 0 \mod p$. Then use the fact that when $yz \equiv 0 \mod p$ then either $y\equiv 0$ or $z \equiv 0 \mod p$.