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Smallest k for m × k to be a perfect cube

Smallest k for m × k to be a perfect cube
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2019-08-14 00:26
Geometric Construction
This example demonstrates how to explore the smallest k such that m × k is a perfect cube using GeoGebra. Core mathematical concepts include prime factorization of composite numbers, GCD, and LCM. The construction utilizes commands and objects like RandomElement, Length, Element, If, Factors, Sequence, GCD, LCM, and Textfield, enabling interactive exploration through sliders and text fields. Suitable for high school mathematics instruction, reviewing concepts of prime factorization and perfect cubes, and fostering students' mathematical inquiry skills.

GeoGebra Commands & Parameters

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

AI Drawing Prompt Reference

Create the smallest factor k such that m×k is a perfect cube. Start by randomly selecting a number c, then find its factors m2. Next, randomly choose a vector A and calculate c1, c2, and c3 based on A's coordinates. Use these values to find the GCD and LCM of m0 and m2. Finally, through factorization and GCD calculations, determine the value of k to ensure m×k is a perfect cube.
The prompt above was compiled from the original drawing commands and is for reference only.