I have an idea of what to do. I know that for $\gcd\left(a,\:10\right)\:=\:a$ to hold true, $a\le 10$ and $10$ also has to be a multiple of $a$.
Therefore, I think the only possible integers are $\pm 1,\:\pm 2,\pm 5,\:\pm 10$ but I'm not quite sure on how to prove this statement.
Any assistance would be appreciated!
To give this question slightly more mathematical content:
Proposition. For all $a, b \in \mathbb{N}$ we have $\operatorname{gcd}(a, b) = a$ if and only if $a \mid b$.
Proof. If $\operatorname{gcd}(a, b) = a$ then in particular $a \mid b$. On the other hand if $a \mid b$ then $\operatorname{gcd}(a, b) \geq a$, but $\operatorname{gcd}(a, b)$ cannot exceed $a$ so we have equality.
(So the answers are the positive factors of $b = 10$.)