Find the length of the plank that can be used to measure the lengths 4 m 20 cm and 5 m 4 cm exactly, in the least time.
Generated by claude-sonnet-4-6 · 2026-06-15 10:30 · grounding rag
Model Answer
Convert to cm: 4 m 20 cm = 420 cm; 5 m 4 cm = 504 cm.
To measure both lengths exactly in least time, the plank must be of maximum possible length, i.e., HCF(420, 504).
Prime factorisation:
- $420 = 2^2 \times 3 \times 5 \times 7$
- $504 = 2^3 \times 3^2 \times 7$
HCF = $2^2 \times 3 \times 7 = **84$ cm**
The length of the plank = 84 cm.
Source: Chapter 1, Section 1.2 (The Fundamental Theorem of Arithmetic)
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Explanation
- "Least time" means fewest number of times the plank is used → plank must be as long as possible → HCF, not LCM.
- Always convert to the same unit before finding HCF.
- Use prime factorisation method: HCF = product of lowest powers of common prime factors.
- Examiners award 1 mark for correct method/factorisation and 1 mark for the correct answer (84 cm).