If ratio and HCF of two numbers are 3 : 4 and 4 respectively. Then find the LCM ?

Options

A. 12
B. 24
C. 36
D. 48

Answer

To find the LCM of two numbers given their ratio and HCF, we can follow these steps:

### Step 1: Understand the given information
We are given the ratio of two numbers as 3:4 and their HCF (Highest Common Factor) as 4.

### Step 2: Set up the numbers based on the ratio
Let the two numbers be represented as:
- First number = 3x
- Second number = 4x

### Step 3: Use the HCF to find the value of x
Since the HCF of the two numbers is given as 4, we can set up the equation:
- HCF(3x, 4x) = x (because the HCF of 3 and 4 is 1, and the common factor is x)
- Therefore, we have x = 4.

### Step 4: Calculate the actual numbers
Now substitute the value of x back into the expressions for the numbers:
- First number = 3x = 3 * 4 = 12
- Second number = 4x = 4 * 4 = 16

### Step 5: Find the LCM of the two numbers
To find the LCM of 12 and 16, we can use prime factorization:
- Prime factorization of 12:
- 12 = 2^2 * 3^1
- Prime factorization of 16:
- 16 = 2^4

### Step 6: Use the highest powers of the prime factors
The LCM is found by taking the highest power of each prime factor from the factorizations:
- For the prime factor 2, the highest power is 2^4 (from 16).
- For the prime factor 3, the highest power is 3^1 (from 12).

### Step 7: Calculate the LCM
Now multiply these highest powers together:
- LCM = 2^4 * 3^1 = 16 * 3 = 48.

### Final Answer
Thus, the LCM of the two numbers is **48**.

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