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Smallest n for m × n to be a perfect square

Smallest n for m × n to be a perfect square
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2019-08-12 11:52
FunctionInequalityGeometric Construction
This example demonstrates how to use GeoGebra to explore the smallest integer n such that m × n is a perfect square. Core mathematical concepts include prime factorization, GCD, and LCM. Construction highlights features like Length, RandomElement, Factors, GCD, LCM, and more. Suitable for high school mathematics review or self-assessment.

GeoGebra Commands & Parameters

  1. Length(l1) → lengthl1
  2. RandomElement({7, 11}) → c
  3. Random(1, lengthl1) → a
  4. Element(l1, a) → A
  5. Factors(b) → m0
  6. Sequence(LaTeX[Element[m0, i, 1] + If[Element[m0, i, 2] ≟ 1, "", "^" + (Element[m0, i, 2])]], i, 1, Length[m0]) → lm0
  7. If(floor(z(A) / 2) - z(A) / 2 ≟ 0, z(A), z(A) + 1) → c3
  8. If(floor(y(A) / 2) - y(A) / 2 ≟ 0, y(A), y(A) + 1) → c2
  9. If(floor(x(A) / 2) - x(A) / 2 ≟ 0, x(A), x(A) + 1) → c1
  10. Factors(c5) → m2
  11. Length(m2) → lengthm2
  12. Sequence(LaTeX["×" + (Element[m2, i, 1]) + (If[Element[m2, i, 2] ≟ 1, "", "^" + (Element[m2, i, 2])])], i, 2, lengthm2) → l4
  13. Length(l4) → lengthl4
  14. If(x(A) ≟ 1, 1, If[x(A) ≟ 2, 0, If[x(A) ≟ 3, 1]]) → d1
  15. If(y(A) ≟ 1, 1, If[y(A) ≟ 2, 0, If[y(A) ≟ 3, 1]]) → d2
  16. If(z(A) ≟ 1, 1, If[z(A) ≟ 2, 0, If[z(A) ≟ 3, 1]]) → d3
  17. Random(1, y(A)) → q2
  18. GCD(b, e1) → e2
  19. LCM(b, e1) → e3
  20. Factors(c4) → m1
  21. Length(m1) → lengthm1
  22. Sequence(LaTeX[Element[m1, i, 1] + "^" + (Element[m1, i, 2])], i, 1, lengthm1) → l2
  23. Sequence(LaTeX[Element[m2, i, 1] + If[Element[m2, i, 2] ≟ 1, "", "^" + (Element[m2, i, 2])]], i, 1, 2) → l3
  24. GCD({c1, c2, c3}) → d
  25. Sequence(LaTeX[Element[m1, i, 1] + If[Element[m1, i, 2] / d ≟ 1, "", "^" + (Element[m1, i, 2] / d)]], i, 1, lengthm1) → l2factored
  26. Textfield(textInput_1) → InputBox1