Two numbers have a LCM of 45. The two numbers add up to 14. Find the two numbers.

To find the two numbers, let's use the given information:

1. The two numbers have a Least Common Multiple (LCM) of 45.
2. The sum of the two numbers is 14.

Let the two numbers be \( x \) and \( y \). Then:

\[
x + y = 14
\]

We also know that the LCM of \( x \) and \( y \) is 45. A key property of the LCM and GCD (Greatest Common Divisor) of two numbers is:

\[
\text{LCM}(x, y) = \frac{x \times y}{\text{GCD}(x, y)}
\]

Since the LCM is 45, we can express this as:

\[
\frac{x \times y}{\text{GCD}(x, y)} = 45
\]

Next, we will check possible pairs of numbers whose LCM is 45 and whose sum is 14.

### Step-by-step calculation:

1. **Factor pairs of 45:**  
   The factor pairs of 45 are:  
   \[
   (1, 45), (3, 15), (5, 9)
   \]

2. We now check which of these pairs has a sum of 14:
   - \( 1 + 45 = 46 \) (doesn't work)
   - \( 3 + 15 = 18 \) (doesn't work)
   - \( 5 + 9 = 14 \) (works!)

Thus, the two numbers are \( 5 \) and \( 9 \).

### Verification:

- The sum is \( 5 + 9 = 14 \), which is correct.
- The LCM of 5 and 9 can be calculated as:
  \[
  \text{LCM}(5, 9) = \frac{5 \times 9}{\text{GCD}(5, 9)} = \frac{45}{1} = 45
  \]

So, the two numbers are indeed \( 5 \) and \( 9 \).

Would you like more details on LCM and GCD or any specific steps?

Here are 5 related questions you might find interesting:
1. What is the LCM of 12 and 18?
2. How do you find the GCD of two numbers using the Euclidean algorithm?
3. What is the sum of two numbers if their LCM is 36 and their product is 144?
4. How can you express the relationship between GCD and LCM using prime factorization?
5. How do you find the LCM of more than two numbers?

**Tip:** The relationship between LCM, GCD, and the product of two numbers is a very useful tool in solving problems like this one. Always consider this formula when given both LCM and GCD.

Solve the problem of finding two numbers that have a Least Common Multiple (LCM) of 45 and a sum of 14. Explore the relationship between LCM and GCD with step-by-step solutions suitable for Grades 6-8.

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