The HCF and LCM of two numbers are 13 and 455 respectively. If one of the numbers lies between 75 and 125, then, that numbers is:

Options

A. a) 78
B. b) 91
C. c) 104
D. d) 117

Answer

To solve the problem, we need to find one of the two numbers given that their HCF is 13 and their LCM is 455. We also know that one of the numbers lies between 75 and 125.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The relationship between the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of two numbers \(a\) and \(b\) is given by:
\[ \text{HCF} \times \text{LCM} = a \times b \]
Here, HCF = 13 and LCM = 455.

2. **Setting Up the Numbers**:
Let the two numbers be \(a = 13x\) and \(b = 13y\), where \(x\) and \(y\) are co-prime integers. This is because the HCF of \(a\) and \(b\) is 13.

3. **Using the Relationship**:
Substitute the values of HCF and LCM into the relationship:
\[ 13 \times 455 = (13x) \times (13y) \]
Simplifying this gives:
\[ 13 \times 455 = 169xy \]
Dividing both sides by 13:
\[ 455 = 13xy \]
Further simplifying gives:
\[ xy = \frac{455}{13} = 35 \]

4. **Finding Pairs of \(x\) and \(y\)**:
We need to find pairs of integers \(x\) and \(y\) such that \(xy = 35\). The pairs are:
- \( (1, 35) \)
- \( (5, 7) \)
- \( (7, 5) \)
- \( (35, 1) \)

5. **Calculating the Corresponding Numbers**:
Now, we calculate the corresponding numbers for each pair:
- For \( (1, 35) \):
- \( a = 13 \times 1 = 13 \)
- \( b = 13 \times 35 = 455 \)
- For \( (5, 7) \):
- \( a = 13 \times 5 = 65 \)
- \( b = 13 \times 7 = 91 \)
- For \( (7, 5) \):
- \( a = 13 \times 7 = 91 \)
- \( b = 13 \times 5 = 65 \)
- For \( (35, 1) \):
- \( a = 13 \times 35 = 455 \)
- \( b = 13 \times 1 = 13 \)

6. **Identifying the Number Between 75 and 125**:
From the calculated pairs, we have:
- \( (13, 455) \) → Not between 75 and 125.
- \( (65, 91) \) → 91 is between 75 and 125.
- \( (91, 65) \) → 91 is between 75 and 125.
- \( (455, 13) \) → Not between 75 and 125.

The only number that lies between 75 and 125 is **91**.

### Conclusion:
The number that lies between 75 and 125 is **91**.