HCF and LCM of two 3 - digit number are 16 and 2640 respectively. Find the numbers?

Options

A. 264, 120
B. 240, 176
C. 300, 160
D. 192, 176

Answer

To find the two 3-digit numbers given their HCF (Highest Common Factor) and LCM (Least Common Multiple), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- HCF = 16
- LCM = 2640

2. **Use the Relationship Between HCF, LCM, and the Numbers:**
- The relationship states that:
\[ \text{HCF} \times \text{LCM} = \text{Number 1} \times \text{Number 2} \]
- Let the two numbers be \(16a\) and \(16b\) (since their HCF is 16).

3. **Set Up the Equation:**
- Substitute the values into the formula:
\[ 16 \times 2640 = (16a) \times (16b) \]
- Simplifying gives:
\[ 16 \times 2640 = 256ab \]
- Dividing both sides by 16:
\[ 2640 = 16ab \]
- Dividing both sides by 16 again:
\[ ab = \frac{2640}{16} = 165 \]

4. **Find Pairs of Factors of 165:**
- The pairs of integers whose product is 165 are:
- (1, 165)
- (3, 55)
- (5, 33)
- (11, 15)

5. **Calculate the Corresponding Numbers:**
- For each pair \((a, b)\), calculate \(16a\) and \(16b\):
- For (1, 165):
- \(16 \times 1 = 16\) (not a 3-digit number)
- \(16 \times 165 = 2640\) (not a 3-digit number)
- For (3, 55):
- \(16 \times 3 = 48\) (not a 3-digit number)
- \(16 \times 55 = 880\) (3-digit number)
- For (5, 33):
- \(16 \times 5 = 80\) (not a 3-digit number)
- \(16 \times 33 = 528\) (3-digit number)
- For (11, 15):
- \(16 \times 11 = 176\) (3-digit number)
- \(16 \times 15 = 240\) (3-digit number)

6. **Identify Valid 3-Digit Numbers:**
- The valid pairs that yield 3-digit numbers are:
- \(176\) and \(240\)

### Final Answer:
The two 3-digit numbers are **176 and 240**.