Math 2.0

Number theory Theorem

Euclidean algorithm step for the greatest common divisor

Builds on Greatest common divisorDivision algorithmDivisibility

Statement

Let mm and nn be positive integers with m>nm > n. Divide mm by nn with the division algorithm. This gives the unique integers qq and rr with m=qn+rm = qn + r and 0≤r<n0 \le r < n.

If r>0r > 0, then gcd⁡(m,n)=gcd⁡(n,r)\gcd(m, n) = \gcd(n, r), where gcd⁡\gcd is the greatest common divisor.

If r=0r = 0, then nn divides mm, and gcd⁡(m,n)=n\gcd(m, n) = n.

Remarks

The first case lets us replace the pair (m,n)(m, n) by the smaller pair (n,r)(n, r) without changing the gcd. The second case ends the process.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionGreatest common divisor, concept, no evidence levelGreatest common divis…conceptDivision algorithm, statement, no evidence levelDivision algorithmstatementEuclidean algorithm step for the greatest common divisor, statement, no evidence levelEuclidean algorithm s…statement
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. Greatest common divisor concept
  3. Division algorithm statement
  4. Euclidean algorithm step for the greatest common divisor statement

Further reading

  1. Web https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_induction_recursion.ptx#L223, source/s_induction_recursion.ptx:223
About this page
Tag9N6JTBDC0N
URIm2://9N6JTBDC0N
Kindstatement, theorem
WritingW1 stated
MSC11A05
StatusThis copy of the site has no build data. Run m2 status 9N6JTBDC0N.
UsesGreatest common divisor, Division algorithm, Divisibility
Changed4 November 2026

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