0580/22 Oct/Nov 2025, Question 13: Prime Factorization
Cambridge IGCSE Mathematics 0580 · Paper 22 (Extended) · Non-calculator · 4 marks
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The question, in short
Given M = 2⁷ × 3³ × 5², express 14M as a product of primes in index form, and find the least integer R for which M ÷ R is a cube.
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Part (a) · 2 marks
Express 14M as a product of prime factors, giving the result using index notation.
- 114 = 2 × 7
- 214M = 2 × 7 × 2⁷ × 3³ × 5² = 2⁸ × 3³ × 5² × 7
Answer2⁸ × 3³ × 5² × 7
Part (b) · 2 marks
Find the smallest integer R such that dividing M by R produces a cube number.
- 1For a cube every index must be a multiple of 3
- 2Divide out 2¹ and 5²: M/R = 2⁶ × 3³ = (2² × 3)³ = 12³
- 3R = 2 × 5² = 50
Answer50
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Also searched as: 0580/22/O/N/25 · 0580_w25_qp_22 · 0580 w25 22 · question 13.
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