Prove $\text{lcm}(x,y)/\text{gcd}(x,y) \in N$

Question

Prove that $$\frac{\text{lcm}(x,y)}{\text{gcd}(x,y)}\in\Bbb N$$ for all $x, y \in \Bbb N$. And what kind of conditions you need, so $\frac{\text{lcm}(x,y)}{\text{gcd}(x,y)}$ would be equal integer's square?

Answer 1

Hint $\,\ d\mid x\mid m\,\Rightarrow\, d\mid m\,$ by transitivity of divisibility.

As for the second question, note that a positive integer is a square iff every prime occurs to even power in its unique prime factorization. But the power of a prime $p$ in the lcm/gcd is even iff its power in $x$ and $y$ have equal parity (since it's their max $-$ min in lcm/gcd). This is equivalent to $x/y$ being a square of a rational, or $xy$ being a square.