Proof problem: If $x-1$ is divisible by $4$ then $x^2 - 1$ is divisible by 4

Question

The question is to state if it's true or false.

Question: If $x-1$ is divisible by 4 then $x^2 - 1$ is divisible by 4

I'm not very familiar with the mathematical language of elementary number theory compared to calculus, algebra as this is mostly calculations and using the correct properties. It would be great if you can show me steps to understand.

Answer 1

If $x−1$ is divisible by $4$ then $\,x -1 = 4n\,$ for some integer $n\,$. Then $\,x^2-1$ $=(4n+1)^2-1$ $\require{cancel}=16n^2 + 8n + \cancel{1} - \cancel{1} = 8n(2n+1)\,$ is divisible by $8$, which is in turn divisible by $4$.

Answer 2

$x^2-1=(x-1)(x+1)$ this implies the result.

Answer 3

If $4\mid x-1$ then for any polynomial $p(x)$ we have $$4\mid (x-1)\cdot p(x)$$

Specially in your case $p(x)=x+1$ and we are done.