Math 2.0

Number theory Proof

Proof of Euclid's lemma

Builds on Prime numberDivisibilityGreatest common divisorCoprime integersGeneralized Euclid's lemma

Proof

We deduce the lemma from the generalized Euclid's lemma.

  1. Let pp be a prime number and let aa, bb be integers with p∣abp \mid ab. If p∣ap \mid a, we are done. So assume p∤ap \nmid a.
  2. Let g=gcd⁡(p,a)g = \gcd(p, a) be the greatest common divisor. It is defined as the largest common divisor, because p≠0p \neq 0.
  3. The number 11 is a common divisor of pp and aa, so g≥1g \ge 1.
  4. The number gg is a positive divisor of pp. Since pp is prime, g=1g = 1 or g=pg = p.
  5. If g=pg = p, then p∣ap \mid a, which is false. So g=1g = 1, and pp and aa are coprime.
  6. By the generalized Euclid's lemma, p∣abp \mid ab and gcd⁡(p,a)=1\gcd(p, a) = 1 give p∣bp \mid b.
  7. So p∣ap \mid a or p∣bp \mid b.

Idea

A prime has only two positive divisors, so a prime that does not divide aa has nothing in common with aa. Then the coprime case of the lemma forces pp to divide bb.

Reading path

To understand this page from the ground up, read these 26 pages in order. The foundations come first. The path takes the proofs Proof of Bézout's identity, Proof of the generalized Euclid's lemma, Proof of the commutativity of addition by induction, Proof of strong induction from ordinary induction, Proof of the well-ordering principle from strong induction, Proof of the division algorithm.

Show the path as a graph
Definition by recursion, concept, no evidence levelDefinition by recursi…conceptDivisibility, definition, no evidence levelDivisibilitydefinitionMathematical induction, concept, no evidence levelMathematical inductionconceptNatural numbers, object, no evidence levelNatural numbersobjectPeano axioms, concept, no evidence levelPeano axiomsconceptGreatest common divisor, definition, no evidence levelGreatest common divis…definitionPrime number, definition, no evidence levelPrime numberdefinitionSuccessor function, definition, no evidence levelSuccessor functiondefinitionCoprime integers, definition, no evidence levelCoprime integersdefinitionAddition of natural numbers, definition, no evidence levelAddition of natural n…definitionThe successor function is injective, statement, no evidence levelThe successor functio…statementZero is not a successor, statement, no evidence levelZero is not a success…statementProof of the commutativity of addition by induction, proof, no evidence levelProof of the commutat…proofOrder on the natural numbers, definition, no evidence levelOrder on the natural …definitionCommutativity of addition, statement, no evidence levelCommutativity of addi…statementProof of strong induction from ordinary induction, proof, no evidence levelProof of strong induc…proofStrong induction, statement, no evidence levelStrong inductionstatementProof of the well-ordering principle from strong induction, proof, no evidence levelProof of the well-ord…proofWell-ordering principle, statement, no evidence levelWell-ordering princip…statementProof of the division algorithm, proof, no evidence levelProof of the division…proofDivision algorithm, statement, no evidence levelDivision algorithmstatementProof of Bézout's identity, proof, no evidence levelProof of Bézout's ide…proofBézout's identity, statement, no evidence levelBézout's identitystatementProof of the generalized Euclid's lemma, proof, no evidence levelProof of the generali…proofGeneralized Euclid's lemma, statement, no evidence levelGeneralized Euclid's …statementProof of Euclid's lemma, proof, no evidence levelProof of Euclid's lem…proof
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
  1. Definition by recursion concept
  2. Divisibility definition
  3. Mathematical induction concept
  4. Natural numbers object
  5. Peano axioms concept
  6. Greatest common divisor definition
  7. Prime number definition
  8. Successor function definition
  9. Coprime integers definition
  10. Addition of natural numbers definition
  11. The successor function is injective statement
  12. Zero is not a successor statement
  13. Proof of the commutativity of addition by induction proof
  14. Order on the natural numbers definition
  15. Commutativity of addition statement
  16. Proof of strong induction from ordinary induction proof
  17. Strong induction statement
  18. Proof of the well-ordering principle from strong induction proof
  19. Well-ordering principle statement
  20. Proof of the division algorithm proof
  21. Division algorithm statement
  22. Proof of Bézout's identity proof
  23. Bézout's identity statement
  24. Proof of the generalized Euclid's lemma proof
  25. Generalized Euclid's lemma statement
  26. Proof of Euclid's lemma proof

Where this leads

175 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Euclid%27s_lemma, Proofs
About this page
Tag85G9HBA7ND
URIm2://85G9HBA7ND
Kindproof
WritingW0 stub
MSC11A05
ProvesEuclid's lemma
UsesPrime number, Divisibility, Greatest common divisor, Coprime integers, Generalized Euclid's lemma
Changed4 November 2026

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