Divisibility by coprime integers

Bézout. $au+bv=1$.

Multiply. $n=n(au+bv)=aun+bvn$.

Divisibility. $a\mid aun$; and $b\mid n$, so $bvn=b\cdot(\text{multiple of }b)\cdot v$. But we need $a\mid bvn$: since $a\mid n$, $n=ka$, so $bvn=bvka=ab(vk)$, which is divisible by $ab$. Similarly $aun=au\cdot kb=ab(uk)$, divisible by $ab$.

Conclusion. $ab\mid n$.