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Factor Sets, Prime Factors, and GCD/LCM of Two Integers

Factor Sets, Prime Factors, and GCD/LCM of Two Integers
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2024-11-30 19:55
SequenceTransformationOptimizationGeometric ConstructionGraphing
In collections Sets
This example lets you input two integers and then visualizes their prime-factor structure by constructing factor sets and comparing shared vs. unique prime powers. It highlights how the greatest common divisor (GCD/HCF) and the least common multiple (LCM) follow from prime exponents. The core math concepts are integer factorization, set-based reasoning about factor collections, frequency/exponent counting in prime decompositions, and the GCD–LCM relationship derived from matching and combining prime powers. Construction-wise, it uses GeoGebra commands such as GCD, LCM, PrimeFactors, Frequency, and Unique to compute prime-factor lists and their exponent multiplicities; it organizes intermediate results with objects like aFact, bFact, onlyaFact, onlybFact, mcdFact, and the corresponding exponent sets (e.g., espOnlyaFact/espmcdFact). InputBox/Textfield objects (InputBox1/InputBox2 and warningText/fullText...) drive updates, while Vector and Sequence help arrange and label the computed outputs. Suitable for middle school through early university, especially for number theory basics and interactive demonstrations of GCD/LCM computation from prime factor sets.

GeoGebra Commands & Parameters

  1. GCD(a, b) → mcdAB
  2. Textfield(a) → InputBox1
  3. Textfield(b) → InputBox2
  4. Point(xAxis) → A
  5. Point(xAxis) → B
  6. Ellipse(A, B, C) → d
  7. Point(xAxis) → D
  8. Point(xAxis) → E
  9. Ellipse(D, E, F) → e
  10. PrimeFactors(a) → aFact
  11. PrimeFactors(b) → bFact
  12. Frequency(aFact) → espaFact
  13. Frequency(bFact) → espbFact
  14. Unique(aFact) → unia
  15. Unique(bFact) → unib
  16. PrimeFactors(a / mcdAB) → onlyaFact
  17. PrimeFactors(b / mcdAB) → onlybFact
  18. PrimeFactors(mcdAB) → mcdFact
  19. Frequency(onlyaFact) → espOnlyaFact
  20. Frequency(onlybFact) → espOnlybFact
  21. Frequency(mcdFact) → espmcdFact
  22. Unique(onlyaFact) → uniOnlya
  23. Unique(onlybFact) → uniOnlyb
  24. Unique(mcdFact) → uniMcd
  25. Vector(H, G) → u
  26. Zip(Text[A], A, espaFact) → fullTextEspa
  27. Zip(Text[A], A, espbFact) → fullTextEspb
  28. Zip(Text[A], A, espOnlybFact) → fullTextEspOnlyb
  29. Zip(Text[A], A, espOnlyaFact) → fullTextEspOnlya
  30. Sequence(If[fullTextEspa(k) ≠ "1", fullTextEspa(k), ""], k, 1, Length[fullTextEspa]) → finalTextEspa
  31. Sequence(If[fullTextEspb(k) ≠ "1", fullTextEspb(k), ""], k, 1, Length[fullTextEspb]) → finalTextEspb
  32. Sequence(If[fullTextEspOnlyb(k) ≠ "1", fullTextEspOnlyb(k), ""], k, 1, Length[fullTextEspOnlyb]) → finalTextEspOnlyb
  33. Sequence(If[fullTextEspOnlya(k) ≠ "1", fullTextEspOnlya(k), ""], k, 1, Length[fullTextEspOnlya]) → finalTextEspOnlya
  34. Zip(Text[A], A, espmcdFact) → fullTextEspmcd
  35. Sequence(If[fullTextEspmcd(k) ≠ "1", fullTextEspmcd(k), ""], k, 1, Length[fullTextEspmcd]) → finalTextEspmcd
  36. LCM(a, b) → mcmAB

AI Drawing Prompt Reference

Input two integers to visualize their prime factorization and factor sets, using ellipses for shared and unique prime power sets, and label the GCD and LCM. Dynamically display prime factor frequencies and exponents via vectors, sequences, and text boxes, emphasizing the relationship between GCD/LCM and prime factors.
The prompt above was compiled from the original drawing commands and is for reference only.