Question

Three numbers are in the ratio of 3 : 4 : 5, and their least common multiple (LCM) is 2400. Their highest common factor (HCF) is

• A. 40
• B. 80
• C. 120
• D. 200
• E. 15

Hint:

If numbers are in a ratio \(a:b:c\) (where \(a,b,c\) are mutually co-prime), their actual values are \(ax, bx, cx\) where \(x\) is the HCF. The LCM is simply the product of their LCM(ratio) and the HCF.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the ratio of three numbers and their Least Common Multiple (LCM). We need to find their Highest Common Factor (HCF).


Step 2: Key Formula or Approach:
Let the three numbers be \(ax\), \(bx\), and \(cx\), where \(a, b, c\) are the parts of the ratio and \(x\) represents their HCF (since the given ratio parts 3, 4, and 5 are co-prime).
The LCM of these numbers is given by \(\text{LCM}(a, b, c) \times x\).


Step 3: Detailed Explanation:
Let the three numbers be \(3x\), \(4x\), and \(5x\). Here, \(x\) is their HCF.
The LCM of the ratio parts 3, 4, and 5 is: \[ \text{LCM}(3, 4, 5) = 60 \] Therefore, the LCM of the numbers \(3x\), \(4x\), and \(5x\) is \(60x\).
We are given that the LCM is 2400. Equating the two: \[ 60x = 2400 \] Solving for \(x\): \[ x = \frac{2400}{60} \] \[ x = 40 \] Since \(x\) represents the HCF, the HCF of the numbers is 40.


Step 4: Final Answer:
The highest common factor (HCF) is 40.