How to divide by negative numbers?

Question

How to divide by negative numbers? Like what is the quotient and remainder when 24 is divided by $-5$ or $-24$ is divided by $5$?

Answer 1

The Division Algorithm states that given a pair of integers $a$ and $b$, with $b \neq 0$, there exist integers $q$ and $r$, with $0 \leq r < |b|$ such that $a = q|b| + r$.

When we divide $24$ by $5$, $q = 4$ and $r = 4$ since $24 = 4 \cdot 5 + 4$.

Since $4 \cdot 5 = (-4)(-5)$, $24 = (-4)(-5) + 4$.

The quotient of $-24$ and $5$ takes a little more thought.

Note that $-24 = -4 \cdot 5 - 4$. Our remainder is negative, but we want $0 \leq r < 5$. Observe that -4 = -5 + 1. Hence, \begin{align*} -24 & = -4 \cdot 5 - 4\\ & = -4 \cdot 5 - 5 + 1\\ & = (-4 - 1)5 + 1\\ & = -5 \cdot 5 + 1 \end{align*} In practice, the following method is more efficient. Note that $-25$ is the largest multiple of $5$ that is less than $-24$ and $-24 - (-25) = 1$. Since $-25 = -5 \cdot 5$, we obtain $$-24 = -25 + 1 = -5 \cdot 5 + 1$$