elementary number theory - $d_1,d_2mid niff [d_1,d_2]mid n,$ [LCM Universal Property]

Is

$$d_1\mid n,d_2\mid n\iff [d_1,d_2]\mid n$$
true ? And if yes, how can I prove it ? I recall that $[d_1,d_2]$ is the least common multiple.

My tries

For the implication : Let $d_1\mid n$ and $d_2\mid n$. Then, $d_1d_2\mid n^2$. By the way, since $d_1,d_2\mid n$, we have that $(d_1,d_2)\mid n$ where $(d_1,d_2)=:\gcd(d_1,d_2)$ and thus

$$[d_1,d_2](d_1,d_2)\mid n^2.$$

Question: How can I get $[d_1,d_2]\mid n$ from this?

For the converse, since $d_1,d_2\mid [d_1,d_2]\mid n$, the claim follow.