Given a system:
$$ \begin{cases} 5x - 3y = 0 \\ \text{lcm}(x, y) = 45 \end{cases} $$
Since, $5x=3y$, I've tried expressing product of $xy$ as:
$$ \text{lcm}(x,y)\gcd(x,y)=xy \\ \gcd(x,y)=\frac{xy}{\text{lcm}(x,y)} $$
I see that $x = 9$ and $y = 15$, I understand that we get it from dividing $\text{lcm}(x,y)$ which is 45 by coefficients, but I don't understand why.
If $\mathrm{lcm}(x, y) = 45$, then we know both $x$ and $y$ are factors of 45, so $x = 3^{a_1} 5^{b_1}$ and $y = 3^{a_2} 5^{b_2}$ with $\max(a_1, a_2) = 2$ and $\max(b_1, b_2) = 1$.
We also know that $5x = 3y$, so $3^{a_1} 5^{b_1 + 1} = 3^{a_2 + 1} 5^{b_2}$. Which then implies that $a_1 > a_2$ and $b_2 > b_1$, forcing both of those to be equal to the maximums, i.e. $a_1 = 2$ and $b_2 = 1$ which is enough to give us the solution.