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

Question

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?

Answer 1

No, there is not a proof of this. This is because LCM is just defined that way, just like how we define operations like multiplication, addition, subtraction, etc.

Answer 2

I think what you are looking for is the following statement:

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 $n\ge c.$