Let $n,m$ be natural numbers with $n\ge 1,$ $m\ge 1,$ and the factorizations $n=\prod_{i=1}^\infty p_i^{e_i}$ and $m=\prod_{i=1}^\infty p_i^{f_i}.$ Then the greatest common divisor $\gcd(n,m)$ and the least common multiple $\operatorname{lcm}(n,m)$ have the following factorizations:
$$\gcd(n,m)=\prod_{i=1}^\infty p_i^{\min(e_i,f_i)},\quad \operatorname{lcm}(n,m)=\prod_{i=1}^\infty p_i^{\max(e_i,f_i)},$$
where $\min(e_i,f_i)$ and $\max(e_i,f_i)$ are the minimum or the maximum of the respective exponents $e_i,f_i.$
Proofs: 1
