The sum of two numbers is 45. Their difference is `(1)/(9)` of their sum. Their L.C.M. is

Options

A. 200
B. 250
C. 100
D. 150

Answer

To solve the problem step by step, we will follow the information given in the question.

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \).

### Step 2: Set Up the Equations
From the problem, we know:
1. The sum of the two numbers is 45:
\[ x + y = 45 \quad \text{(Equation 1)} \]
2. The difference of the two numbers is \( \frac{1}{9} \) of their sum:
\[ x - y = \frac{1}{9} \times 45 \quad \text{(Equation 2)} \]
Simplifying Equation 2:
\[ x - y = 5 \]

### Step 3: Solve the Equations
Now we have two equations:
1. \( x + y = 45 \)
2. \( x - y = 5 \)

We can solve these equations by adding them together:
\[ (x + y) + (x - y) = 45 + 5 \]
This simplifies to:
\[ 2x = 50 \]
Dividing both sides by 2:
\[ x = 25 \]

### Step 4: Find the Value of \( y \)
Now that we have \( x \), we can find \( y \) using Equation 1:
\[ y = 45 - x = 45 - 25 = 20 \]

### Step 5: Find the LCM of \( x \) and \( y \)
Now we have \( x = 25 \) and \( y = 20 \). To find the LCM, we can use the prime factorization method:
- The prime factorization of \( 25 \) is \( 5^2 \).
- The prime factorization of \( 20 \) is \( 2^2 \times 5^1 \).

To find the LCM, we take the highest power of each prime factor:
- For \( 2 \): \( 2^2 \)
- For \( 5 \): \( 5^2 \)

Thus, the LCM is:
\[ LCM = 2^2 \times 5^2 = 4 \times 25 = 100 \]

### Final Answer
The LCM of the two numbers is \( 100 \).

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