how to prove lcm divides any common multiple from definition of lcm(a,b) = (a*b)/gcd(a,b)

Question

So the goal is to prove that the $\operatorname{lcm}(a,b)$ divides any multiple of of $a$ and $b$. Suppose there is some integer $c$ such that $a|c$ and $b|c$ but I want to prove $\operatorname{lcm}(a,b)|c$ also. I got that $$\operatorname{lcm}(a,b)=\frac{(a\cdot b)}{\gcd(a,b)}$$ and I want to see how we could show $$\frac{(a*b)}{\gcd(a,b)}\bigg| \, c$$ Any help would be appreciated thank you.

Answer 1

WARNING: This answer is INCORRECT. See comments for what mistakes I am making

Firstly we know that some multiple of $a$ and $b$ can be represented by $kab$ where $k$ is some integer, and thus is divisible by $ab$.

We need to show that $lcm(a,b) \vert ab$. Since $gcd(a,b)$ would be some integer, and that $lcm(a,b) gcd(a,b) = ab$, it is obvious.

Therefore any multiple of $ab$ is divisible by $lcm(a,b)$.