The sum of two numbers is 45. Their difference is `1/9` of their sum. Their LCM is:

Options

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

Answer

To solve the problem step-by-step, we will follow the information provided in the question and the video transcript.

### Step 1: Define the two numbers
Let the two numbers be \( N_1 \) and \( N_2 \).

### Step 2: Set up the equations based on the problem statement
From the problem, we know:
1. The sum of the two numbers is 45:
\[ N_1 + N_2 = 45 \quad \text{(1)} \]
2. The difference between the two numbers is \( \frac{1}{9} \) of their sum:
\[ N_1 - N_2 = \frac{1}{9} \times 45 \quad \text{(2)} \]

### Step 3: Calculate the difference
Calculating \( \frac{1}{9} \times 45 \):
\[ \frac{1}{9} \times 45 = 5 \]
So, we can rewrite equation (2) as:
\[ N_1 - N_2 = 5 \quad \text{(3)} \]

### Step 4: Solve the equations
Now we have two equations:
1. \( N_1 + N_2 = 45 \) (1)
2. \( N_1 - N_2 = 5 \) (3)

We can add equations (1) and (3):
\[ (N_1 + N_2) + (N_1 - N_2) = 45 + 5 \]
This simplifies to:
\[ 2N_1 = 50 \]
Dividing both sides by 2 gives:
\[ N_1 = 25 \]

### Step 5: Find \( N_2 \)
Now that we have \( N_1 \), we can find \( N_2 \) using equation (1):
\[ 25 + N_2 = 45 \]
Subtracting 25 from both sides gives:
\[ N_2 = 20 \]

### Step 6: Find the LCM of \( N_1 \) and \( N_2 \)
Now we need to find the LCM of 25 and 20.

**Finding the prime factors:**
- The prime factorization of 25 is:
\[ 25 = 5^2 \]
- The prime factorization of 20 is:
\[ 20 = 2^2 \times 5^1 \]

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

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

### Final Answer:
The LCM of the two numbers is \( \boxed{100} \).

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