Whilst answering my homework, I have noticed that a question has said that I need to consider the subgroups of $A=\mathbb{Z}_{135}$, including $W=\langle 18,81\rangle$. I was under the impression that for a subset to be a subgroup $a+b$ in the subgroup had to be in the subgroup as well. $18+81 = 99\mod{135}$ which isn't in the subgroup.
Is it my reasoning or the problem sheet that is wrong? Thank you in advance.