If x=103, then the LCM of `x^(2)-4` and `x^(2)-5x+6` is

Options

A. 105105
B. 1051050
C. 106050
D. 1060500

Answer

To find the LCM of the expressions \(x^2 - 4\) and \(x^2 - 5x + 6\) when \(x = 103\), we can follow these steps:

### Step 1: Factor the expressions
1. **Factor \(x^2 - 4\)**:
- This is a difference of squares, which can be factored as:
\[ x^2 - 4 = (x - 2)(x + 2) \]

2. **Factor \(x^2 - 5x + 6\)**:
- We look for two numbers that multiply to \(6\) (the constant term) and add to \(-5\) (the coefficient of \(x\)). The numbers \(-2\) and \(-3\) work:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 2: Identify the LCM
3. **Identify the LCM**:
- The LCM is found by taking the highest power of each factor from both factorizations:
- From \(x^2 - 4\): \( (x - 2) \) and \( (x + 2) \)
- From \(x^2 - 5x + 6\): \( (x - 2) \) and \( (x - 3) \)
- The LCM is:
\[ \text{LCM} = (x - 2)(x + 2)(x - 3) \]

### Step 3: Substitute \(x = 103\)
4. **Substitute \(x = 103\) into the LCM**:
- Calculate each term:
\[ (103 - 2) = 101, \quad (103 + 2) = 105, \quad (103 - 3) = 100 \]
- Therefore, the LCM becomes:
\[ \text{LCM} = 101 \times 105 \times 100 \]

### Step 4: Calculate the product
5. **Calculate the product**:
- First, calculate \(101 \times 105\):
\[ 101 \times 105 = 10605 \]
- Now multiply by \(100\):
\[ 10605 \times 100 = 1060500 \]

### Final Answer
The LCM of \(x^2 - 4\) and \(x^2 - 5x + 6\) when \(x = 103\) is:
\[ \text{LCM} = 1060500 \]