Question

Show that if a ≡ b (mod n) and m divides n, then a ≡ b (mod m).

Answer 1

If $a\equiv b\mod n$, then there is some integer $N$ such that $a=Nn+b$. Because $m|n$, it follows that $m|Nn$, which means there is some integer $M$ such that $a=Mm+b$, which is equivalent to the statement that $a\equiv b\mod m$.