Find, with proof, all the integers $a$ that satisfy the equation $\gcd\left(a,\:10\right)\:=\:a.$ [duplicate]
Solution 1:
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 $b = 10$.)