Proving or disproving gcd and lcm

Question

I'm having a bit of trouble with this type of problem:

$∀a, b ∈ Z^+, gcd(a, b) = gcd(ab, lcm(a, b))$

How would you expand the RHS to eventually equal to the LHS or vice versa? And ultimately what is the correct way to prove this statement?

Answer 1

Since $\operatorname{lcm}(a,b)$ is necessarily a factor of $ab,$ then $$\operatorname{gcd}\bigl(ab,\operatorname{lcm}(a,b)\bigr)=\operatorname{lcm}(a,b).$$ Thus, the claim amounts to saying that $$\operatorname{gcd}(a,b)=\operatorname{lcm}(a,b)$$ for all positive integers $a,b$. This will only be true when $a=b,$ however.

I highly recommend that, in the future, you work several examples to get a feel for general statements you're supposed to prove or disprove. You may find a counterexample (which, of course, disproves the general statement), or you may get a good feel for why it's true and how to prove it.