Greatest measuring length (HCF application): Find the greatest possible length (in centimetres) that can exactly measure each of the following three lengths: 4 m 3 cm, 4 m 34 cm, and 4 m 65 cm.

Question

Solve this multiple-choice question and choose the correct option.

Accepted Answer

Introduction / Context:This is a classic Highest Common Factor (HCF) or Greatest Common Divisor (GCD) problem framed as measuring equal segments. The greatest length that measures several lengths exactly is the HCF of those lengths when all are in the same unit. Given Data / Assumptions: Lengths: 4 m 3 cm, 4 m 34 cm, 4 m 65 cm. Convert to a single unit (centimetres) to compute the HCF. Goal: greatest length that exactly divides all three. Concept / Approach:When a rod or tape is to measure several lengths exactly, the required length is the HCF. Converting metres to centimetres simplifies arithmetic. Also, when numbers form an arithmetic progression, their HCF must divide the common difference, which can simplify checks. Step-by-Step Solution: Convert units: 4 m 3 cm = 403 cm; 4 m 34 cm = 434 cm; 4 m 65 cm = 465 cm.Observe differences: 434 − 403 = 31; 465 − 434 = 31.Since the three numbers are in arithmetic progression with common difference 31, the HCF divides 31.Check divisibility by 31: 403/31 = 13, 434/31 = 14, 465/31 = 15 (all integers).Therefore HCF = 31 cm. Verification / Alternative check: Compute HCF(403, 434) = 31, then HCF(31, 465) = 31, confirming the result. Why Other Options Are Wrong: 29 cm, 28 cm, 32 cm do not divide all three lengths 403 cm, 434 cm, and 465 cm without remainder. Common Pitfalls: Forgetting to convert metres to centimetres consistently. Taking LCM by mistake instead of HCF. Final Answer: 31 cm

Greatest measuring length (HCF application): Find the greatest possible length (in centimetres) that can exactly measure each of the following three lengths: 4 m 3 cm, 4 m 34 cm, and 4 m 65 cm.

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