Math 2.0

Number theory Proof

Proof that the lcm of coprime integers is their product

Builds on Coprime integersLeast common multipleProduct of the gcd and the lcm

Proof

  1. Let aa and bb be natural numbers that are coprime. By definition, gcd⁡(a,b)=1\gcd(a, b) = 1.
  2. By the product of the gcd and the lcm, gcd⁡(a,b)⋅lcm⁡(a,b)=ab\gcd(a, b) \cdot \operatorname{lcm}(a, b) = a b.
  3. Substituting gcd⁡(a,b)=1\gcd(a, b) = 1 gives lcm⁡(a,b)=ab\operatorname{lcm}(a, b) = a b, where lcm⁡\operatorname{lcm} is the least common multiple.

Idea

The gcd times the lcm is always the product. When the gcd is 11, nothing is left to divide out, so the lcm is the product itself.

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 commutativity of addition by induction, Proof of the product of the gcd and the lcm, 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…definitionLeast common multiple, definition, no evidence levelLeast common multipledefinitionSuccessor 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 product of the gcd and the lcm, proof, no evidence levelProof of the product …proofProduct of the gcd and the lcm, statement, no evidence levelProduct of the gcd an…statementProof that the lcm of coprime integers is their product, proof, no evidence levelProof that the lcm of…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. Least common multiple 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 product of the gcd and the lcm proof
  25. Product of the gcd and the lcm statement
  26. Proof that the lcm of coprime integers is their product proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Least_common_multiple
About this page
Tag1TRB7CW647
URIm2://1TRB7CW647
Kindproof
WritingW0 stub
MSC11A05
ProvesLcm of coprime integers
UsesCoprime integers, Least common multiple, Product of the gcd and the lcm
Changed4 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.