Question about how they get forms in proof of "The square of any odd int has the form 8m+1 for some int m"

Question

Can someone tell me how they determine that it can be written in those particular forms?

Answer 1

Notice that for any integer $x$, the remainder of $x$ when divided by $4$ has to be $0,1,2$ or $3$. Let $r$ be this remainder. Since $x-r$ must be divisible by $4$, we can let $\frac{x-r}{4} = q$ for some integer $q$. Thus:

$$ \begin{align*} x & = 4(\frac{x-r+r}{4})\\ & = 4(\frac{x-r}{4} + \frac{r}{4})\\ & = 4q + r \end{align*} $$

Since $r$ is $0,1,2$ or $3$, then $x$ has to be of the form $4q$,$4q+1$,$4q+2$ or $4q+3$