If both x and y are integers, how many solutions are there of the equation `(x-8)(x-10) = 2y` ?

Options

A. 1
B. 2
C. 3
D. more than 3

Answer

To find the integer solutions for the equation \((x-8)(x-10) = 2y\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ (x-8)(x-10) = 2y \]
This means that the product of \((x-8)\) and \((x-10)\) must be even since it equals \(2y\).

### Step 2: Analyze the left-hand side
The left-hand side can be expanded:
\[ x^2 - 18x + 80 = 2y \]

### Step 3: Determine conditions for \(y\)
Since \(2y\) is even, \((x-8)(x-10)\) must also be even. This occurs if at least one of the factors \((x-8)\) or \((x-10)\) is even.

### Step 4: Identify integer values for \(x\)
Now we will check the values of \(x\) that make \((x-8)(x-10)\) even.

1. **Case 1**: \(x\) is even.
- If \(x\) is even, both \(x-8\) and \(x-10\) are even. Thus, \((x-8)(x-10)\) is even.

2. **Case 2**: \(x\) is odd.
- If \(x\) is odd, both \(x-8\) and \(x-10\) are odd. Thus, \((x-8)(x-10)\) is odd, which cannot equal \(2y\).

### Step 5: Find integer solutions
Since \(x\) must be even, let’s denote \(x = 2k\) where \(k\) is an integer. Then we can substitute \(x\) back into the equation:
\[ (2k-8)(2k-10) = 2y \]
This simplifies to:
\[ (2(k-4))(2(k-5)) = 2y \]
or
\[ 4(k-4)(k-5) = 2y \]
Dividing both sides by 2, we get:
\[ 2(k-4)(k-5) = y \]

### Step 6: Count integer solutions for \(k\)
The expression \(2(k-4)(k-5)\) can take integer values based on the integer values of \(k\). We can find the integer pairs \((k-4)\) and \((k-5)\) that yield integer results.

1. Set \(k-4 = a\) and \(k-5 = a-1\):
\[ y = 2a(a-1) \]
This means \(y\) can be any integer that results from this formula.

### Step 7: Identify specific integer solutions
Now we can find specific integer values for \(x\) (even integers) and calculate corresponding \(y\):
- For \(x = 10\):
- \((10-8)(10-10) = 2y \Rightarrow 2 \cdot 0 = 2y \Rightarrow y = 0\)
- For \(x = 12\):
- \((12-8)(12-10) = 2y \Rightarrow 4 \cdot 2 = 2y \Rightarrow y = 4\)

### Final Count of Solutions
Thus, we found two integer solutions:
1. \( (10, 0) \)
2. \( (12, 4) \)

### Conclusion
The total number of integer solutions \((x, y)\) for the equation \((x-8)(x-10) = 2y\) is **2**.

---