1. ## division algorithm problem

i'm not sure how to even approach this problem. any help?

Suppose $\displaystyle a,b,q,r e Z$ such that $\displaystyle a=bq+r$, $\displaystyle b>0$ and $\displaystyle 0<=r<b$. then $\displaystyle q=[a/b]$ and $\displaystyle r=a-b[a/b]$

(Hint. Use the uniqueness of the division algorithm).

Suppose $\displaystyle a,b,q,r e Z$ such that $\displaystyle a=bq+r$, $\displaystyle b>0$ and $\displaystyle 0<=r<b$. then $\displaystyle q=[a/b]$ and $\displaystyle r=a-b[a/b]$
$\displaystyle a=bq+r \implies \frac{a}{b} = q + \frac{r}{b}$.Since $\displaystyle 0 \leq r < b$, $\displaystyle q = \left[\frac{a}{b}\right]$