Doubt regarding cyclic group of prime power order

Question

Let $G=\langle x,y\rangle$ be a cyclic group of prime power order. Then is it true that $G$ is generated by either $x$ or $y$?

What we tried:

Let $G=\langle z\rangle.$ Then $x=z^m, y=z^n$ for some integer $m,n.$ So $G=\langle x,y\rangle=\langle z^{gcd(m,n)}\rangle=\langle z\rangle.$ This gives $gcd(m,n)=1.$ By given condition, the order of $z$ is $p^r,$ for some prime $p$ and integer $r.$ Need to show either $m$ or $n$ is 1.

Answer 1

Hint: It is not true, consider e.g. $G=C_6, x=z^2, y=z^3$.

EDIT: Oops, $G$ is of prime power order. It may help to write $m=kp^i$ and $n=lp^j$ with $k,l\nmid p$. What can you say about $i,j$?

Answer 2

You might write $G=\mathbb{Z}/p^n\mathbb{Z}$, $(x)=p^k \mathbb{Z}/p^n\mathbb{Z}$, $(y)=p^m\mathbb{Z}/p^n\mathbb{Z}$. Then, assume without loss of generality that $k \geq m$ (otherwise, interchange the roles of $x$ and $y$). It follows that $(x) \subset (y)$, so $(x,y)=(y)$. The result follows from setting $G=(x,y)$.