Q: Let $G=\mathbb{Z}_{25}$ and $H=\langle 13\rangle$ be the subgroup of $G$ generated by $13$. Yes or No: do $24$ and $23$ belong to the same $H$-coset in $G$?
No; my reasoning: Know that $2$ elements belong to the same coset if they are the same equivalence class and $\gcd(25,13)=1$, so 13 will generate the entire group $\mathbb{Z}_{25}$. $a \sim_Hb\iff -a+b\in H\implies23\sim_H24\iff-23+24\in H\implies1\in H \;\square$
Is my line of reasoning valid? How else could I have easily answer this "yes/no" question?