Nash Frank

2022-07-16

Discrete math: proofs
For any three integers x, y, and z, if 𝒚 is divisible by x and z is divisible by y, then z is divisible by x.
definition: An integer n is divisible by an integer d with $D\ne 0$, denoted d|n, if and only if there exists an integer k such that $n=dk$.
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?

kitskjeja

Expert

Step 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 $b\in \mathbb{Z}$.
Step 2
Substituting in the first equation, $z=a\left(bx\right)=\left(ab\right)x$, and $ab\in \mathbb{Z}$, so z is divisible by x.

Ibrahim Rosales

Expert

Step 1
If y is divisible by x, we have $y=kx$ for $k\in \mathbb{Z}-\left\{0\right\}$. And if z is divisible by y, we have $z=my$ for $m\in \mathbb{Z}-\left\{0\right\}$.
Step 2
Now, putting $y=kx$ to $z=my$, we have $z=mkx$. Therefore by definition, x|z.

Do you have a similar question?