Showing that $\operatorname{lcm}(a,b)$ is the unique positive generator of $(a) \cap (b)$

Question

Let $a,b \in \mathbb{N}$. $l>0$ is the unique positive generator of the ideal $(a) \cap (b)$. Show that $l = \frac{ab}{d}$ where $d = gcd(a, b)$. I am stuck on this problem. $(a)=\{na: n \in \mathbb{Z}\}$...

Answer 1

Hint 1: Show that $l \in (a) \cap (b)$.

Hint 2: Show that no proper divisor of $l$ can be in $(a) \cap (b)$.