The greatest possible length which can exactly measure both a length of 80…

Concept: The greatest length that can measure two given lengths an exact (whole) number of times is the Highest Common Factor (HCF), i.e. the Greatest Common Divisor (GCD), of the two lengths. The HCF is the product of the common prime factors taken to their lowest powers, so it is the largest length that divides both exactly.

Application: First express both lengths in the same unit (cm), then take their HCF.

1. Convert to cm: 80 meter 64 cm = (80 × 100) + 64 = 8000 + 64 = 8064 cm.

2. Convert to cm: 96 meter = 96 × 100 = 9600 cm.

3. Factorise 8064 = 27 × 32 × 7.

4. Factorise 9600 = 27 × 3 × 52.

5. Common prime factors at lowest powers = 27 × 3 = 128 × 3 = 384.

Result: HCF = 384, so the greatest possible measuring length is 384 cm.

Cross-check: 8064 ÷ 384 = 21 and 9600 ÷ 384 = 25 — both exact, with no common factor between 21 and 25, confirming 384 is the largest such length.