Congruences with LCM and Relatively Prime Numbers

Question

How do I verify that if $a \equiv b\pmod{n_1}$ and $a \equiv b\pmod{n_2}$, then $a \equiv b \pmod n$, where the integer $n = \operatorname{lcm} (n_1, n_2)$. Hence, whenever $n_1$ & $n_2$ are relatively prime, $a \equiv b \pmod{n_1 n_2}$