$n_1,...,n_k$ pair coprime $\!\iff\! {\rm lcm}(n_1,...,n_k)=n_1...n_k$ [lcm = product for coprimes] $n_1,...,n_k$ pairwise coprime $\iff LCM(n_1,...,n_k)=n_1...n_k$
Recently, I was told this as part of a larger proof concerning direct products of groups. I am wondering why this is true. 

Can you prove the following two statements?


*

*If $a \mid c$ and $b \mid c$ with $\gcd(a,b) = 1$ then $ab \mid c$.

*If $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$ then $gcd(a,bc) = 1$.


Then, use induction to show that $n_1 \dots n_k$ divides $\text{lcm}(n_1,\dots,n_k)$. Since $n_1\dots n_k$ is a common multiple that divides the least common multiple, it must be equal to the least common multiple.
