Find the greatest possible length which can be used to measure exactly the…

Concept: The greatest length that can measure two or more given lengths exactly is the Highest Common Factor (HCF) of those lengths expressed in the same unit — any length that divides all the given lengths exactly must be a common factor of all of them, and the largest such common measure is precisely their HCF.

1. Convert all lengths to a common unit (cm): 4 m 95 cm = 495 cm, 9 m = 900 cm, 16 m 65 cm = 1665 cm.

2. Write each as a product of prime factors: 495 = 32 × 5 × 11, 900 = 22 × 32 × 52, 1665 = 32 × 5 × 37.

3. Take the common prime factors at their lowest powers across all three numbers: only 32 and 51 are common to all three, so HCF = 32 × 5 = 45.

Cross-check: 495 ÷ 45 = 11, 900 ÷ 45 = 20, and 1665 ÷ 45 = 37 — all exact integers, confirming that 45 cm measures every one of the three lengths exactly. Since the only prime factors shared by all three numbers are 32 and 5, no larger common measure exists.

Result: the greatest possible length that measures all three lengths exactly is 45 cm.