Division theorem for positive integers

Statement

Take two positive integers $m$ and $n$ . Then exactly one pair of integers $(q, r)$ satisfies both $m = qn + r$ and $0 \le r < n$ .

Remarks

The integer $q$ is the quotient and $r$ is the remainder when $m$ is divided by $n$ . The remainder is $0$ exactly when $n$ divides $m$ . For example, with $m = 17$ and $n = 5$ we get $q = 3$ and $r = 2$ , since $17 = 3 \cdot 5 + 2$ . This is the case of the general division algorithm in which both numbers are positive.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Divisibility definition
Division theorem for positive integers statement

Tag	`5DD48BF6CM`
URI	`m2://5DD48BF6CM`
Aliases	`division-theorem-positive-integers`
Kind	statement, theorem
Writing	W1 stated
MSC	11A05
Status	This copy of the site has no build data. Run `m2 status 5DD48BF6CM`.
Uses	Divisibility
Changed	4 November 2026

Division theorem for positive integers

Statement

Remarks

Reading path

Further reading

What this page needs for W2

What links here