The product of LCM and HCF of two numbers is 63504. If one of the numbers is 378 then find the HCF of numbers?

Options

A. 21
B. 42
C. 63
D. 56

Answer

To find the HCF of the two numbers given that the product of their LCM and HCF is 63504, and one of the numbers is 378, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between LCM, HCF, and the two numbers**:
The relationship states that:
\[ \text{LCM} \times \text{HCF} = \text{Product of the two numbers} \]
Let's denote the two numbers as \( a \) and \( b \). Here, \( a = 378 \) and we need to find \( b \) and the HCF.

2. **Set up the equation**:
We know:
\[ \text{LCM} \times \text{HCF} = 63504 \]
Hence, we can express it as:
\[ \text{HCF} \times \text{LCM} = 378 \times b \]

3. **Find the second number \( b \)**:
Rearranging the equation gives us:
\[ b = \frac{63504}{378} \]
Now, we will calculate \( b \):
\[ b = 168 \]

4. **Factorize both numbers**:
Now, we need to factorize both 378 and 168 to find their HCF.
- Factorization of 378:
\[ 378 = 2 \times 3^3 \times 7 \]
- Factorization of 168:
\[ 168 = 2^3 \times 3 \times 7 \]

5. **Identify common factors**:
To find the HCF, we take the lowest power of all common prime factors:
- For \( 2 \): The minimum power is \( 2^1 \)
- For \( 3 \): The minimum power is \( 3^1 \)
- For \( 7 \): The minimum power is \( 7^1 \)

6. **Calculate the HCF**:
Now we multiply these common factors:
\[ \text{HCF} = 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 = 42 \]

### Final Answer:
The HCF of the two numbers is **42**.