Prove $\,\gcd(a,b)\mid {\rm lcm}(a,b)\ $

Question

I have a hard time with this question, can someone explain?

"Prove that for any number $a$ and $b$, $(a,b)$ goes into $\text{lcm}(a;b)$"

Answer 1

Hint: by gcd & lcm universal property $\,\gcd(a,b)\mid a\mid {\rm lcm}(a,b)\,$ so by transitivity of divisibility ...

Remark $ $ The same argument works for any number of arguments to gcd & lcm. The hint boils down to $\,\inf S \le \sup S\,$ in poset / lattice language. More generally, the hint shows that every common divisor of a set of integers divides every common multiple.