Let $\displaystyle G=\mathbb{Z}_4\times\mathbb{Z}_3$. Show that $\displaystyle G$ is isomorphic to $\displaystyle \mathbb{Z}_{12}$.
Take $\displaystyle (1,1)\in \mathbb{Z}_4 \times \mathbb{Z}_3$ then let $\displaystyle k\in \mathbb{N}$ and we have $\displaystyle (k1,k1)=0$ iff $\displaystyle 4\vert k$ and $\displaystyle 3\vert k$ the least such number is $\displaystyle [4,3]$ (the min. comm. mult.) and it's well known that $\displaystyle [a,b](a,b)=ab$ then since $\displaystyle (4,3)=1$...