Question

There are three ropes of lengths 4 m 50 cm, 9 m 90 cm and 16 m 20 cm, respectively. Each rope can be used to exactly measure the side of a square ground. What is the maximum possible value of each side, in m, if area of the square ground is less than 10,00,000 m\(^2\)?

• A. 891
• B. 781
• C. 791
• D. 871

Hint:

When finding the LCM of large numbers, factor out common multiples like 10 or 100 first to simplify the prime factorization step.

Answer 1

The Correct Option is A

Step 1: Convert Lengths to Common Unit (cm):
Rope 1: 4 m 50 cm = 450 cm 
Rope 2: 9 m 90 cm = 990 cm 
Rope 3: 16 m 20 cm = 1620 cm 

Step 2: Find LCM of Rope Lengths: 
The side of the square must be a multiple of all rope lengths. \[ 450 = 10 \times 45 = 2 \times 3^2 \times 5^2 \] \[ 990 = 10 \times 99 = 2 \times 3^2 \times 5 \times 11 \] \[ 1620 = 10 \times 162 = 2^2 \times 3^4 \times 5 \] \(\text{LCM} = 2^2 \times 3^4 \times 5^2 \times 11\) \(\text{LCM} = 4 \times 81 \times 25 \times 11 = 100 \times 891 = 89100 \text{ cm} = 891 \text{ m}\). 

Step 3: Check Area Constraint: 
Area \(< 10,00,000 \text{ m}^2\). Side \(< \sqrt{1,000,000} = 1000 \text{ m}\). The LCM is 891 m, which is less than 1000 m. The next multiple would be \(891 \times 2 = 1782\) m, which exceeds the limit. Therefore, the maximum side is 891 m.