# How do you prove something is the LCM of two numbers?

I am wondering if there is a proof validating the definition of the LCM. I know that the least common multiple of two integers, say $a, b$ is just the smaller number $n$, such that $a|n$ and $b|n$, but is there a proof that goes along with this?

• If you want to check that $n$ is indeed $LCM(a,b)$, compute $GCD(n/a,n/b)$. It must be $1$.
– user65203
Oct 21 '15 at 21:15
• This should be an answer. Oct 21 '15 at 21:17

Let $a,b\in\mathbb N$. Let $c = \frac{ab}{(a,b)} =: \operatorname{lcm}(a,b).$ Then:
• $c$ is a natural number
• $a|c$ and $b|c$
• For all $n,$ if $a|n$ and $b|n$ then $n\ge c.$
• In fact, you can also replace $n\ge c$ with $c|n$ and the result will still hold Oct 23 '15 at 8:43