Math 2.0

Number theory Theorem

Division algorithm

Builds on Divisibility

Statement

Let aa and bb be integers with b≠0b \neq 0. Then there is exactly one pair of integers (q,r)(q, r) such that a=bq+ra = b q + r and 0≤r<∣b∣0 \le r < |b|.

Remarks

The integer qq is the quotient and the integer rr is the remainder of the division of aa by bb. The remainder is 00 if and only if bb divides aa. For example, −7=3⋅(−3)+2-7 = 3 \cdot (-3) + 2, so the quotient of −7-7 by 33 is −3-3 and the remainder is 22.

Reading path

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

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Divisibility, definition, no evidence levelDivisibilitydefinitionDivision algorithm, statement, no evidence levelDivision algorithmstatement
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.
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  1. Divisibility definition
  2. Division algorithm statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Euclidean_division, Division theorem
About this page
Tag0CHQKZFK5S
URIm2://0CHQKZFK5S
Aliaseseuclidean-division, division-with-remainder
Kindstatement, theorem
WritingW0 stub
MSC11A05
StatusThis copy of the site has no build data. Run m2 status 0CHQKZFK5S.
UsesDivisibility
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