LCM of two numbers is 1260 and their difference is 54. Then find the sum of these two numbers?

Options

A. 88
B. 306
C. 216
D. 342

Answer

To solve the problem, we need to find two numbers whose LCM is 1260 and whose difference is 54. We will then calculate the sum of these two numbers.

### Step-by-Step Solution:

1. **Understanding the Problem**: We know that the LCM of two numbers is 1260 and their difference is 54. Let's denote the two numbers as \( x \) and \( y \) where \( x > y \). Therefore, we can write:
\[ x - y = 54 \]
and
\[ \text{LCM}(x, y) = 1260. \]

2. **Finding Possible Pairs**: To find \( x \) and \( y \), we can look for pairs of factors of 1260. Here are some pairs:
- \( 1 \times 1260 \)
- \( 2 \times 630 \)
- \( 3 \times 420 \)
- \( 4 \times 315 \)
- \( 5 \times 252 \)
- \( 6 \times 210 \)
- \( 7 \times 180 \)
- \( 9 \times 140 \)
- \( 10 \times 126 \)
- \( 12 \times 105 \)
- \( 14 \times 90 \)
- \( 15 \times 84 \)
- \( 18 \times 70 \)
- \( 20 \times 63 \)
- \( 21 \times 60 \)
- \( 28 \times 45 \)
- \( 30 \times 42 \)
- \( 35 \times 36 \)

3. **Checking the Difference**: We need to check which of these pairs have a difference of 54.
- For \( (18, 70) \):
\[ 70 - 18 = 52 \quad \text{(not a match)} \]
- For \( (21, 60) \):
\[ 60 - 21 = 39 \quad \text{(not a match)} \]
- For \( (30, 42) \):
\[ 42 - 30 = 12 \quad \text{(not a match)} \]
- For \( (36, 35) \):
\[ 36 - 35 = 1 \quad \text{(not a match)} \]
- For \( (54, 126) \):
\[ 126 - 54 = 72 \quad \text{(not a match)} \]
- For \( (63, 20) \):
\[ 63 - 20 = 43 \quad \text{(not a match)} \]
- For \( (70, 18) \):
\[ 70 - 18 = 52 \quad \text{(not a match)} \]

After checking several pairs, we find that \( (63, 9) \) gives:
\[ 63 - 9 = 54 \quad \text{(this is a match)} \]

4. **Calculating the Sum**: Now that we have found the two numbers \( x = 63 \) and \( y = 9 \), we can calculate their sum:
\[ x + y = 63 + 9 = 72. \]

5. **Final Answer**: Therefore, the sum of the two numbers is:
\[ \boxed{72}. \]