How do I show that there are integers $x, y$ with $[a, b]$ = $ax + by$

Question

I've seen the answer for gcd$(a,b)$ but never for the lcm$(a,b)$?

Answer 1

For lcm$(a,b)$, it's a multiple of $a$, so we can take $x=\frac{lcm(a,b)}{a}, y=0$. The gcd case, as you can see, is more interesting.

Answer 2

Well this actually follows from the answer for the gcd. Here's a big hint, suppose x' and y' are such that gcd(a,b) = ax'+by'. Now, we know the gcd divides both a and b and that lcm is divisible by both a and b. So try working from there toward a solution.