Three numbers are in the ratio 2 : 3 : 4 and their H.C.F. is 12 . The L.C.M. of the numbers is

Options

A. 144
B. 192
C. 96
D. 72

Answer

To find the LCM of three numbers that are in the ratio 2:3:4 and have an HCF of 12, we can follow these steps:

### Step 1: Understand the relationship between HCF and the numbers
Since the HCF of the three numbers is 12, we can express the numbers in terms of their HCF. Let the three numbers be:
- First number = 12 * 2 = 24
- Second number = 12 * 3 = 36
- Third number = 12 * 4 = 48

### Step 2: Write the numbers
Now we have the three numbers:
- First number = 24
- Second number = 36
- Third number = 48

### Step 3: Find the LCM of the numbers
To find the LCM of 24, 36, and 48, we can use the prime factorization method.

**Prime Factorization:**
- 24 = 2^3 * 3^1
- 36 = 2^2 * 3^2
- 48 = 2^4 * 3^1

### Step 4: Determine the highest powers of each prime factor
Now, we take the highest power of each prime factor from the factorizations:
- For 2, the highest power is 2^4 (from 48).
- For 3, the highest power is 3^2 (from 36).

### Step 5: Calculate the LCM
Now, we can calculate the LCM using these highest powers:
\[ \text{LCM} = 2^4 * 3^2 \]
Calculating this gives:
\[ \text{LCM} = 16 * 9 = 144 \]

### Final Answer
Thus, the LCM of the numbers is **144**.

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