Discrete math: proofs

Question

For any three integers , , and , if is divisible by and is divisible by , then is divisible by .

definition: An integer is divisible by an integer with ≠0, denoted | , if and only if there exists an integer such that =.

Can anyone help, with this problem I don't know how to approach it should I use proof by cases where every number is odd or even. Should I use direct proof or indirect proof?

Answer 1

If $z$ is divisible by $y$, then $z=ay$ for some $a\in \mathbb{Z}$.

Now, $y$ is divisible by $x$, so $y=bx$ for some $b\in \mathbb{Z}$.

Substituing in the first equation, $z=a(bx)=(ab)x$, and $ab\in \mathbb{Z}$, so $z$ is divisible by $x$.

Answer 2

If $y$ is divisible by $x$, we have $y = kx$ for $k \in \mathbb{Z}-\{0\}$. And if $z$ is divisible by $y$, we have $z = my$ for $m \in \mathbb{Z}-\{0\}$. Now, putting $y = kx$ to $z = my$, we have $z = mkx$. Therefore by definition, $x|z$.