Proving a subgroup is cyclic

Question

Let $G=Z_n$ for $n\gt1$ and let $a,b \in G$ where $a,b$ are two integers (with at least one nonzero). Prove that the subgroup of G generated by $a$ and $b$ is indeed cyclic and is generated by $c \in Z_n$ where $c=gcd(a,b)$.

I know that anything in this subgroup is congruent to 0 mod a and mod b, so they have to be congruent to gcd(ab), I guess I'm just not really sure how to show this properly.

Answer 1

We know that the additive group $Z_n$ is cyclic since the element $1 \in Z_n$ generates all $Z_n $ for every $n \in \Bbb N $

Then we just need to use that every subgroup of a cyclic group is also cyclic. Have you seen this proof yet?

Answer 2

Euclidean algorithm gives $$c = ax+by$$ for some integers $x$ and $y$. So $c$ is in the subgroup generated by $a$ and $b$. On the other hand, as you said, everything in that subgroup is congruent to $0$ modulo $c$.