Quotient Remainder Theorem (QRT)

Division

Let us review concepts and terms you learned as a newborn, starting with division.

Long Division Form

$$\begin{array}{r}quotient\phantom{)}\\ divisor\overline{)dividend\phantom{)}}\end{array}$$

Fractional Form

$$quotient=\frac{dividend}{divisor}$$

Abbreviations

$d=divisor$, $q=quotient$, $n=dividend$, $r=remainder$

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Why Use the Letter $n$ for $dividend$?

We can use any letter except $d$ since $divisor$ already has it. Maybe $n$ is used because $dividend$ occupies the same space as the $numerator$ does in fractions.

$$\frac{\underset{―}{n}umerator}{denominator}$$

The same letters are used to represent the Quotient-Remainder Theorem.

$$n=dq+r$$$$dividend=(divisor)(quotient)+remainder$$

QRT

Given any integer $n$, and a positive integer $d$, there exists unique integers $q$ and $r$ such that:

$$n=dq+rwhere0\le r<d$$

Furthermore, $q$ and $r$ are uniquely determined by $n$ and $d$.

One significant takeway here is that $r$ must always be positive.

Citations, References, and Links