Difficulty: Easy
Correct Answer: 0.18
Explanation:
Introduction:
This question checks your ability to handle greatest common divisor (GCD or HCF) calculations when numbers are given in decimal form. The key idea is to convert the decimals into whole numbers, find the HCF, and then convert back if necessary.
Given Data / Assumptions:
Concept / Approach:
Dealing directly with decimals in divisibility problems is inconvenient. Instead, multiply all numbers by 100 to remove the decimal points. This does not change their ratio. Then find the HCF of the resulting integers. Finally, divide that HCF by 100 to restore the original scale and get the GCD of the decimal numbers.
Step-by-Step Solution:
Multiply each number by 100 to avoid decimals: 1.08 × 100 = 108 0.36 × 100 = 36 0.90 × 100 = 90 Now find HCF of 108, 36, and 90. First find HCF(108, 36): 108 = 2^2 × 3^3 36 = 2^2 × 3^2 So HCF(108, 36) = 2^2 × 3^2 = 4 × 9 = 36 Now find HCF(36, 90): 90 = 2 × 3^2 × 5 Common primes: 2^1 × 3^2 = 2 × 9 = 18 So HCF(108, 36, 90) = 18 We had multiplied by 100, so divide by 100: GCD of original decimals = 18 ÷ 100 = 0.18
Verification / Alternative check:
You can check that 0.18 divides each of the original numbers exactly: 1.08 ÷ 0.18 = 6, 0.36 ÷ 0.18 = 2, and 0.90 ÷ 0.18 = 5. No larger decimal will divide all three numbers without leaving a remainder, so 0.18 is indeed the greatest common divisor.
Why Other Options Are Wrong:
0.03 and 0.108 are smaller common divisors but not the greatest. 0.90 is larger than some of the numbers and clearly cannot divide all of them. Therefore only 0.18 satisfies the definition of GCD for these decimals.
Common Pitfalls:
Students sometimes forget to scale all numbers by the same power of 10, or they attempt to find the HCF directly on decimals, which can be confusing. Remember that multiplying by a common factor and later dividing by the same factor preserves divisibility relationships.
Final Answer:
The greatest common divisor of 1.08, 0.36, and 0.90 is 0.18.
Discussion & Comments