Divisibility proof and its reverse implication

Question

Suppose we have that prove that $mk|n \implies m|n \text{ and } k|n$. This is my idea so far. Since by definition of divisibility $m$ divides $n$ if there exists $q$ such that $n = mq$. Let $q = k$ (from assumption), then $m|n$. I can use similar argument for $k|n$ case. Is my reasoning correct? Also would the reverse implication be true? I think it would be false since i can use same arguement and show that even if $m|n$ and $k|n$ there is a chance that $mk > n$ and thus it wont be true. An example would be 2|4 and 4|4 but $(2\cdot 4) |4$ is not true