The product of HCF and LCM of two numbers is 3321. If one of the numbers is 369, the HCF of the numbers is:

Options

A. 3
B. 21
C. 9
D. 27

Answer

To solve the problem step by step, we need to find the HCF of two numbers given that the product of their HCF and LCM is 3321 and one of the numbers is 369.

### Step 1: Understand the relationship between HCF, LCM, and the numbers
The relationship between the HCF (Highest Common Factor), LCM (Lowest Common Multiple), and the two numbers can be expressed as:
\[ \text{HCF} \times \text{LCM} = \text{Number 1} \times \text{Number 2} \]
Given that one of the numbers (Number 1) is 369, we can denote the other number as \( b \).

### Step 2: Set up the equation
From the relationship, we can write:
\[ \text{HCF} \times \text{LCM} = 369 \times b \]
We also know from the problem statement that:
\[ \text{HCF} \times \text{LCM} = 3321 \]
Thus, we can equate the two expressions:
\[ 369 \times b = 3321 \]

### Step 3: Solve for \( b \)
To find \( b \), we can rearrange the equation:
\[ b = \frac{3321}{369} \]
Now, we can perform the division:
\[ b = 9 \]

### Step 4: Find the HCF
Now that we have both numbers (369 and 9), we can find the HCF of these two numbers. The HCF of 369 and 9 can be calculated as follows:
- The prime factorization of 369 is \( 3^2 \times 41 \).
- The prime factorization of 9 is \( 3^2 \).

The HCF is the product of the lowest powers of all prime factors present in both numbers:
\[ \text{HCF}(369, 9) = 3^2 = 9 \]

### Conclusion
Thus, the HCF of the two numbers is:
\[ \text{HCF} = 9 \]