If $a \mid c$ and $b \mid c$ where $a, b, c \in \mathbb{N}$, under what conditions does it follow that $a \mid b$?

Question

The following question is pretty basic, and the underlying idea was used in the "proof" of a statement in this hyperlinked answer to another MSE question.

The question is as follows:

If $a \mid c$ and $b \mid c$ where $a, b, c \in \mathbb{N}$, under what conditions does it follow that $a \mid b$?

MY ATTEMPT

Take $c = 20$. $c$ factors as follows: $$c = 20 = 4 \cdot 5 = 2 \cdot 10.$$

Note that we can take $a = 2$, $b = 10$. And also note the counterexample $$a = 4 \nmid 5 = b.$$

So I think a condition under which $$\bigg(a \mid c \text{ and } b \mid c\bigg) \implies a \mid b$$ is when $$\frac{b}{a} \mid c.$$ But that condition is too artificial for my purposes. Are there other more natural conditions?

Answer 1

$c=1$ is the only possibility.

Indeed, if $c\neq 1$, then $c \text{ } | \text{ }c$ and $1 \text{ } | \text{ } c$, but obviously you don't have $c \text{ } | \text{ } 1$.