I could use some help with this proof.
Let $n, m ∈ Z^+$ and $a, b ∈ Z$. Suppose that $ a ≡ b$(mod n) and $a ≡ b$(mod m) and $(m, n) = 1.$ Show that $a ≡ b$(mod mn).
From what I understand, it is obvious that $n|(a-b)$ and $m|(a-b)$. So since $(m, n) = 1$ does it mean that there is some prime factor that divides $mn$ like so: $p^a | mn$? Can someone help me figure this proof out, I'm not totally sure if I'm doing this right.
Any help is greatly appreciated! Thanks!