The HCF and LCM of 77, 99 and x are 11 and 3465 respectively. The minimum value of x is

Options

A. 11
B. 55
C. 45
D. 35

Answer

To find the minimum value of \( x \) given that the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 77, 99, and \( x \) are 11 and 3465 respectively, we can follow these steps:

### Step 1: Identify the prime factorization of the given numbers
- The prime factorization of \( 77 \) is \( 7 \times 11 \).
- The prime factorization of \( 99 \) is \( 9 \times 11 = 3^2 \times 11 \).

### Step 2: Write down the HCF and LCM conditions
- The HCF of \( 77, 99, \) and \( x \) is given as \( 11 \). This indicates that \( x \) must also have \( 11 \) as a factor.
- The LCM of \( 77, 99, \) and \( x \) is given as \( 3465 \).

### Step 3: Find the prime factorization of the LCM
To find the prime factorization of \( 3465 \):
- Divide \( 3465 \) by \( 5 \): \( 3465 \div 5 = 693 \).
- Divide \( 693 \) by \( 3 \): \( 693 \div 3 = 231 \).
- Divide \( 231 \) by \( 3 \): \( 231 \div 3 = 77 \).
- We already know \( 77 = 7 \times 11 \).

Thus, the prime factorization of \( 3465 \) is:
\[ 3465 = 3^2 \times 5 \times 7 \times 11 \]

### Step 4: Set up the equation for \( x \)
Since \( x \) must include \( 11 \) and the LCM must account for all prime factors:
\[ \text{LCM}(77, 99, x) = 3^2 \times 5 \times 7 \times 11 \]
We already have \( 77 = 7 \times 11 \) and \( 99 = 3^2 \times 11 \). Therefore, \( x \) must contain the prime factors \( 5 \) and any additional factors needed to ensure the LCM is correct.

### Step 5: Determine the minimum value of \( x \)
Since \( x \) must contain \( 5 \) (to meet the LCM requirement) and must also include \( 11 \) (to meet the HCF requirement), the minimum value of \( x \) can be:
\[ x = 5 \times 11 = 55 \]

### Conclusion
Thus, the minimum value of \( x \) is \( 55 \).

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