To solve the problem, we need to find how many straight lines can be formed that pass through the point (4, 3), have an x-intercept that is a prime number, and a y-intercept that is a positive integer.
### Step-by-Step Solution:
1. **Write the equation of the line in intercept form:**
The intercept form of the equation of a line is given by:
\[
\frac{x}{p} + \frac{y}{q} = 1
\]
where \( p \) is the x-intercept and \( q \) is the y-intercept.
2. **Substituting the point (4, 3) into the equation:**
Since the line passes through the point (4, 3), we substitute \( x = 4 \) and \( y = 3 \) into the equation:
\[
\frac{4}{p} + \frac{3}{q} = 1
\]
3. **Rearranging the equation:**
Rearranging the equation gives:
\[
\frac{4}{p} = 1 - \frac{3}{q}
\]
This can be rewritten as:
\[
\frac{3}{q} = 1 - \frac{4}{p}
\]
4. **Finding a common denominator:**
We can express the right side with a common denominator:
\[
\frac{3}{q} = \frac{p - 4}{p}
\]
5. **Cross-multiplying:**
Cross-multiplying gives us:
\[
3p = q(p - 4)
\]
Rearranging this, we find:
\[
q = \frac{3p}{p - 4}
\]
6. **Identifying conditions for \( q \):**
Since \( q \) must be a positive integer, \( p - 4 \) must divide \( 3p \). This means \( p - 4 \) must be a divisor of \( 12 \) (since \( 3p \) can be expressed as \( 3(p - 4) + 12 \)).
7. **Finding divisors of 12:**
The positive divisors of 12 are \( 1, 2, 3, 4, 6, 12 \). Therefore, we can set:
\[
p - 4 = d \quad \text{where } d \text{ is a divisor of } 12
\]
Hence, \( p = d + 4 \).
8. **Calculating possible values of \( p \):**
- If \( d = 1 \), then \( p = 5 \)
- If \( d = 2 \), then \( p = 6 \) (not prime)
- If \( d = 3 \), then \( p = 7 \)
- If \( d = 4 \), then \( p = 8 \) (not prime)
- If \( d = 6 \), then \( p = 10 \) (not prime)
- If \( d = 12 \), then \( p = 16 \) (not prime)
9. **Identifying prime numbers:**
The only prime numbers obtained from the above calculations are \( p = 5 \) and \( p = 7 \).
10. **Conclusion:**
Therefore, there are **two** straight lines that satisfy the conditions of the problem.
### Final Answer:
The number of straight lines is **2**.
