Proof Writing · Hard problem

Assume $a \mid b$ and $b \mid c$ and $a \nmid c$, and derive a contradiction.

Assume $a \nmid b$ or $b \nmid c$ and $a \mid c$, and derive a contradiction.

Assume $a \mid c$ and show $a \mid b$ and $b \mid c$.

Assume $a \nmid c$ and show $a \mid c$.