Suppose that $a, b, c$ and $d$ are positive integers such that $b$ is an integer multiple of $a$, and $d$ is an integer multiple of $c$. How can we prove that
if the direct sums $ \mathbb Z_a\oplus \mathbb Z_b $ and $\mathbb Z_c\oplus \mathbb Z_d $ are isomorphic then $a=c$ and $b=d$.
What I have done is:
If $b$ is multiple of $a$, then there exists an integer $m$ such that $ b=a\cdot m $. Similarly, if $d$ is an integer multiple of $c$, there exists an integer $n$ such that $ d=c\cdot n $
If $ \mathbb Z_a\oplus \mathbb Z_b $ and $\mathbb Z_c\oplus \mathbb Z_d $ are isomorphic, then $ a\cdot b=c\cdot d $
Then we get $ a^2\cdot m= c^2\cdot n $ . But it seems like we can not get anything from this to reach the answer.