Let $f : \mathbb{Z}_8 \rightarrow \mathbb{Z}_2 \times \mathbb{Z}_4$ be defined by $f([m]_8)=([m]_2,[m]_4)$ for $m \in \mathbb{Z}$ (assuming that $f$ is well defined). Find integers $a$ and $b$ such that $f([a]_8) = f([b]_8)$, but $[a]_8 \neq [b]_8$
In order to show that $[a]_8 \neq [b]_8$ it suffices to show that $a \neq b$ mod $8$
Perhaps I am misunderstanding, but is this question simply just asking to find (as one example)
$ 30 \not\equiv 7$ mod $8$ where $a = 30, $and $b$ = 7
How about $a=0$ and $b=4$? For these numbers, $f([a]_8)=f([b]_8)=(0,0)$ but $[a]_8=0\ne4=[b]_8$.