Find the number of positive integer solutions of the equation `2/x+(15)/(y)= 5`.

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the number of positive integer solutions of the equation \( \frac{2}{x} + \frac{15}{y} = 5 \), we will follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ \frac{2}{x} + \frac{15}{y} = 5 \]
We can rearrange this to isolate \( \frac{15}{y} \):
\[ \frac{15}{y} = 5 - \frac{2}{x} \]

### Step 2: Express \( y \) in terms of \( x \)
Now, we can express \( y \) in terms of \( x \):
\[ y = \frac{15}{5 - \frac{2}{x}} \]
To simplify this, we can find a common denominator:
\[ y = \frac{15x}{5x - 2} \]

### Step 3: Ensure \( y \) is a positive integer
For \( y \) to be a positive integer, \( 5x - 2 \) must be a divisor of \( 15x \). This means \( 5x - 2 \) must divide \( 15x \).

### Step 4: Find integer values of \( x \)
We will check for positive integer values of \( x \) such that \( 5x - 2 \) divides \( 15x \).

1. **For \( x = 1 \)**:
\[ y = \frac{15 \cdot 1}{5 \cdot 1 - 2} = \frac{15}{3} = 5 \]
So, one solution is \( (1, 5) \).

2. **For \( x = 2 \)**:
\[ y = \frac{15 \cdot 2}{5 \cdot 2 - 2} = \frac{30}{8} = 3.75 \]
Not an integer.

3. **For \( x = 3 \)**:
\[ y = \frac{15 \cdot 3}{5 \cdot 3 - 2} = \frac{45}{13} \approx 3.46 \]
Not an integer.

4. **For \( x = 4 \)**:
\[ y = \frac{15 \cdot 4}{5 \cdot 4 - 2} = \frac{60}{18} \approx 3.33 \]
Not an integer.

5. **For \( x = 5 \)**:
\[ y = \frac{15 \cdot 5}{5 \cdot 5 - 2} = \frac{75}{23} \approx 3.26 \]
Not an integer.

6. **For \( x = 6 \)**:
\[ y = \frac{15 \cdot 6}{5 \cdot 6 - 2} = \frac{90}{28} \approx 3.21 \]
Not an integer.

7. **For \( x = 7 \)**:
\[ y = \frac{15 \cdot 7}{5 \cdot 7 - 2} = \frac{105}{33} \approx 3.18 \]
Not an integer.

8. **For \( x = 8 \)**:
\[ y = \frac{15 \cdot 8}{5 \cdot 8 - 2} = \frac{120}{38} \approx 3.16 \]
Not an integer.

9. **For \( x = 9 \)**:
\[ y = \frac{15 \cdot 9}{5 \cdot 9 - 2} = \frac{135}{43} \approx 3.14 \]
Not an integer.

10. **For \( x = 10 \)**:
\[ y = \frac{15 \cdot 10}{5 \cdot 10 - 2} = \frac{150}{48} \approx 3.125 \]
Not an integer.

### Conclusion
After checking all values from \( x = 1 \) to \( x = 10 \), we find that the only positive integer solution is \( (1, 5) \).

Thus, the number of positive integer solutions is **1**.

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