Find the number of positive integers x satisfying the equation `1/x + 1/(x+1) + 1/(x+2) = 13/12`.

Answer

To solve the equation \( \frac{1}{x} + \frac{1}{x+1} + \frac{1}{x+2} = \frac{13}{12} \), we will follow these steps:

### Step 1: Combine the fractions on the left side
We start by finding a common denominator for the fractions on the left side. The common denominator for \( x \), \( x+1 \), and \( x+2 \) is \( x(x+1)(x+2) \).

\[ \frac{(x+1)(x+2) + x(x+2) + x(x+1)}{x(x+1)(x+2)} = \frac{13}{12} \]

### Step 2: Expand the numerator
Now we will expand the numerator:

\[ (x+1)(x+2) = x^2 + 3x + 2 \]
\[ x(x+2) = x^2 + 2x \]
\[ x(x+1) = x^2 + x \]

Adding these together:

\[ x^2 + 3x + 2 + x^2 + 2x + x^2 + x = 3x^2 + 6x + 2 \]

So, we have:

\[ \frac{3x^2 + 6x + 2}{x(x+1)(x+2)} = \frac{13}{12} \]

### Step 3: Cross-multiply
Now we will cross-multiply to eliminate the fractions:

\[ 12(3x^2 + 6x + 2) = 13x(x+1)(x+2) \]

### Step 4: Expand both sides
Expanding the left side:

\[ 36x^2 + 72x + 24 \]

Expanding the right side:

\[ 13x(x^2 + 3x + 2) = 13x^3 + 39x^2 + 26x \]

### Step 5: Set the equation to zero
Now we set the equation to zero by moving all terms to one side:

\[ 36x^2 + 72x + 24 - (13x^3 + 39x^2 + 26x) = 0 \]

This simplifies to:

\[ -13x^3 + (36x^2 - 39x^2) + (72x - 26x) + 24 = 0 \]

\[ -13x^3 - 3x^2 + 46x + 24 = 0 \]

### Step 6: Multiply through by -1
To make the leading coefficient positive, we multiply through by -1:

\[ 13x^3 + 3x^2 - 46x - 24 = 0 \]

### Step 7: Use the Rational Root Theorem
Now we can use the Rational Root Theorem to find possible rational roots. The possible rational roots are the factors of -24 divided by the factors of 13. The factors of -24 are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24, and the factors of 13 are ±1, ±13.

### Step 8: Test possible roots
We will test positive integer values for \( x \):

1. **For \( x = 1 \)**:
\[ 13(1)^3 + 3(1)^2 - 46(1) - 24 = 13 + 3 - 46 - 24 = -54 \quad (\text{not a root}) \]

2. **For \( x = 2 \)**:
\[ 13(2)^3 + 3(2)^2 - 46(2) - 24 = 13(8) + 3(4) - 92 - 24 = 104 + 12 - 92 - 24 = 0 \quad (\text{is a root}) \]

3. **For \( x = 3 \)**:
\[ 13(3)^3 + 3(3)^2 - 46(3) - 24 = 13(27) + 3(9) - 138 - 24 = 351 + 27 - 138 - 24 = 216 \quad (\text{not a root}) \]

4. **For \( x = 4 \)**:
\[ 13(4)^3 + 3(4)^2 - 46(4) - 24 = 13(64) + 3(16) - 184 - 24 = 832 + 48 - 184 - 24 = 672 \quad (\text{not a root}) \]

### Conclusion
The only positive integer solution we found is \( x = 2 \).

### Final Answer
The number of positive integers \( x \) satisfying the equation is **1** (specifically, \( x = 2 \)).