Show that < Z(mod2) X Z(mod3), +> is a cyclic group. The Z's have 2 and 3 has subscripts...
Hint - Lemma: if $\displaystyle x,y$ are two elements of finite coprime order in a group and they commute (i.e., $\displaystyle xy=yx$), then $\displaystyle ord(xy)=ord(x)ord(y)$