Let $G$ be finite Abelian group of order $n_1n_2\cdots n_r$. Let both $G$ and $\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\times \cdots \times \mathbb{Z}_{n_r}$ contain $\delta_1, \delta_2, \cdots, \delta_r$ number of elements of order $n_1, n_2, \cdots, n_r$ respectively. Does it mean $G$ is isomorphic to $\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\times \cdots \times \mathbb{Z}_{n_r}$?
I believe it is true but proving it in rigourous manner find little hard. Neither I could get any counter example. Any kind of help will be appreciated.