How would I go about to prove this?

Question

$n\in \{0,1,2,3,4,5,6,7,8,9\}$

Prove that $n$ is evenly divided by $5$ only if $n$ is $5$ or $0$.

Only way I could think of proving it would be repeat like this:

$0\mod 5 = 0$

$1\mod 5 = 1$

etc... But it doesn't feel acceptable.

Answer 1

You could say that $5$ divides $n$ means that there exists an integer $k$ such that

$$n=5k.$$

• For $k<0$, you will have $n=5k<0$ so $n$ will not belong to your set.

• For $k=0$, $n=0$ which works.

• For $k=1$, $n=5$ which works.

• For $k>1$, $n\geqslant 10$, so $n$ will not belong to your set.