Shortcut Trick
Convert all lengths to cm: 600, 270, and 1120.
The greatest measuring length is the HCF of these values.
By inspection, all numbers end in 0, so 10 is a common factor.
Remaining values: 60, 27, and 112.
Since 27 (33) and 112 (24 × 7) have no common prime factors, the HCF is 10.
∴ The correct answer is 10 cm.
Alternate Method
Given:
Length 1 = 6 m = 600 cm
Length 2 = 2 m 70 cm = 270 cm
Length 3 = 11 m 20 cm = 1120 cm
Formula: Greatest possible length = HCF (Highest Common Factor) of all given lengths.
Calculations:
⇒ Prime factorization of 600 = 2 × 2 × 2 × 3 × 5 × 5 = 23 × 3 × 52
⇒ Prime factorization of 270 = 2 × 3 × 3 × 3 × 5 = 2 × 33 × 5
⇒ Prime factorization of 1120 = 2 × 2 × 2 × 2 × 2 × 5 × 7 = 25 × 5 × 7
⇒ Common prime factors are 2 and 5.
⇒ HCF = 21 × 51
⇒ HCF = 10 cm
∴ The correct answer is 10 cm.
Additional Information
Highest Common Factor (HCF)
It is the greatest number that divides two or more numbers without leaving a remainder. For measuring problems, the HCF represents the largest common scale.
Unit Standardization
Always convert all given quantities into the same unit (usually the smallest unit present) before calculating HCF or LCM (1 m = 100 cm).
HCF of Three Numbers
HCF(a, b, c) = HCF(HCF(a, b), c). This property allows breaking down complex calculations into simpler pairs.