The number of positive integer pairs (a, b) such that ab - 24 = 2a is

Answer

To find the number of positive integer pairs \((a, b)\) such that \(ab - 24 = 2a\), we can follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ ab - 24 = 2a \]
Rearranging it gives:
\[ ab - 2a = 24 \]

### Step 2: Factoring the Left-Hand Side
We can factor the left-hand side:
\[ a(b - 2) = 24 \]

### Step 3: Finding Positive Integer Solutions
For \(a(b - 2) = 24\) to hold true, both \(a\) and \(b - 2\) must be positive integers. This means \(a\) must be a divisor of 24. We will find the positive divisors of 24.

### Step 4: Listing the Divisors of 24
The positive divisors of 24 are:
1. \(1\)
2. \(2\)
3. \(3\)
4. \(4\)
5. \(6\)
6. \(8\)
7. \(12\)
8. \(24\)

### Step 5: Finding Corresponding Values of \(b\)
For each divisor \(a\), we can find \(b\) using the equation:
\[ b - 2 = \frac{24}{a} \implies b = \frac{24}{a} + 2 \]

Now we will calculate \(b\) for each divisor \(a\):

1. If \(a = 1\):
\[ b = \frac{24}{1} + 2 = 24 + 2 = 26 \]

2. If \(a = 2\):
\[ b = \frac{24}{2} + 2 = 12 + 2 = 14 \]

3. If \(a = 3\):
\[ b = \frac{24}{3} + 2 = 8 + 2 = 10 \]

4. If \(a = 4\):
\[ b = \frac{24}{4} + 2 = 6 + 2 = 8 \]

5. If \(a = 6\):
\[ b = \frac{24}{6} + 2 = 4 + 2 = 6 \]

6. If \(a = 8\):
\[ b = \frac{24}{8} + 2 = 3 + 2 = 5 \]

7. If \(a = 12\):
\[ b = \frac{24}{12} + 2 = 2 + 2 = 4 \]

8. If \(a = 24\):
\[ b = \frac{24}{24} + 2 = 1 + 2 = 3 \]

### Step 6: Listing the Valid Pairs
The valid pairs \((a, b)\) are:
1. \((1, 26)\)
2. \((2, 14)\)
3. \((3, 10)\)
4. \((4, 8)\)
5. \((6, 6)\)
6. \((8, 5)\)
7. \((12, 4)\)
8. \((24, 3)\)

### Conclusion
Thus, the number of positive integer pairs \((a, b)\) that satisfy the equation is **8**.

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