Three numbers are in the ratio of 3 : 4 : 5 and their L.C.M. is 4200 . Their H.C.F is

Options

A. 60
B. 20
C. 70
D. 15

Answer

To solve the problem step by step, we will determine the H.C.F of three numbers that are in the ratio of 3:4:5 and have an L.C.M of 4200.

### Step-by-Step Solution:

1. **Understanding the Ratio**:
The three numbers are in the ratio of 3:4:5. We can express these numbers in terms of a variable \( k \):
- First number = \( 3k \)
- Second number = \( 4k \)
- Third number = \( 5k \)

2. **Finding the L.C.M**:
The L.C.M of the numbers \( 3k, 4k, \) and \( 5k \) can be calculated as follows:
- The L.C.M of the coefficients (3, 4, and 5) is found by taking the highest powers of all prime factors involved:
- Prime factorization:
- \( 3 = 3^1 \)
- \( 4 = 2^2 \)
- \( 5 = 5^1 \)
- Therefore, the L.C.M is:
\[ \text{L.C.M} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]
- Hence, the L.C.M of \( 3k, 4k, 5k \) is:
\[ \text{L.C.M} = 60k \]

3. **Setting up the Equation**:
We know from the problem statement that the L.C.M is 4200. Therefore, we can set up the equation:
\[ 60k = 4200 \]

4. **Solving for \( k \)**:
To find \( k \), we divide both sides of the equation by 60:
\[ k = \frac{4200}{60} \]
Simplifying this gives:
\[ k = 70 \]

5. **Finding the H.C.F**:
The H.C.F of the numbers \( 3k, 4k, 5k \) is simply \( k \) because 3, 4, and 5 have no common factors other than 1. Therefore:
\[ \text{H.C.F} = k = 70 \]

### Final Answer:
The H.C.F of the three numbers is **70**.

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