Prove $a,b \in \mathbb{Z}; p, q \in \mathbb{Z^+}$: If $a \equiv b \pmod{pq}\space \longrightarrow \space a \equiv b \pmod p$ $pq|a-b$ and $p|a-b$
$pql = a-b$ and $pk=a-b$
Let $k=ql$ then,
$pql = pk=a-b$
First, is this correct? Second, if so, why can we let $k=ql$?  How do we know $ql=k$?

Hint. If $pq$ divides $a-b$, then $p$ divides $a-b$. Then...

Alternatively, you set $q\ell=\frac{a-b}{p}$ and $k=\frac{a-b}{p}$, so they must be equal.
