Common divisors and common multiples

Common divisors. Let $d=\divisibility(a,b)$ and let $c$ be any common divisor of $a,b$. Write $d=au+bv$ (Bézout). Since $c\mid a$ and $c\mid b$, we get $c\mid au+bv=d$.

Common multiples. Let $M=\operatorname{lcm}(a,b)$ and let $N$ be any common multiple. Write the Euclidean division $N=Mq+r$ with $0\le r<M$. Since $N$ and $M$ are both multiples of $a$ and $b$, so is $r=N-Mq$. But $M$ is the *smallest* positive common multiple, so $r=0$. Hence $M\mid N$.