Let $a = −215$ and $b = 17$. Find the integers $q$ and $r$ with $0 \leq r < b$ such that $a = qb+r$.

Question

Let $a = −215$ and $b = 17$. Find the integers $q$ and $r$ with $0 \leq r < b$ such that $a = qb+r$.

I don't know where to stop

$$a = qb + r$$ $$-215 = q \cdot (-17) + r$$

help me continue.

thanks

Answer 1

You want a $q$ so that $q17$ is smaller thant $-215$, and you want the largest $q$ that achieves this. Notice $\frac{-125}{17}=-7.35$ approximately. So take $q$ to be $-8$. This way we get $-8(17)=-126$. So now $r$ is going to be $1$.

Answer 2

a simple calculation gives $$-215=(-13)\cdot 17+6$$