# Quotient

The quotient is the result of a division, so $a/b$ is called the quotient of $a$ by $b$.

Sometimes, one is interested in the number of times one thing can be divided ‘exactly’ into another, giving a quotient and a remainder.

For example, when working in the integers, $13$ divided by $3$ has a quotient of $4$ and a remainder of $1$.

As another example, $x^4+x^2-x$ divided by $x^2+3$ has a quotient of $x^2-2$ and a remainder of $-x+6$.

If $a$ divided by $b$ has a quotient of $q$ and a remainder of $r$, then $a=bq+r$. Normally, one requires $r$ to be ‘smaller’ than $b$ in some sense. When dividing integers, we require $0\le r<|b|$. When dividing polynomials, we require $r=0$ or the degree of $r$ to be less than the degree of $b$.

The remainder of a division is $0$ if and only if $b$ is a factor of $a$.