To solve the problem, we need to find the number of points with prime number coordinates that lie inside the triangle formed by the lines \(xy = 0\) and \(x + y = 17\) in the x-y plane.
### Step-by-Step Solution:
1. **Identify the Triangle Vertices**:
- The line \(xy = 0\) corresponds to the x-axis and y-axis. This means the triangle is bounded by the x-axis, y-axis, and the line \(x + y = 17\).
- The vertices of the triangle are:
- \(A(0, 0)\) (origin)
- \(B(17, 0)\) (intersection of \(x + y = 17\) with x-axis)
- \(C(0, 17)\) (intersection of \(x + y = 17\) with y-axis)
2. **Equation of the Line**:
- The line \(x + y = 17\) can be rewritten as \(y = 17 - x\). This line intersects the axes at points \(B\) and \(C\).
3. **Determine the Area of the Triangle**:
- The area of triangle \(ABC\) can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 17 \times 17 = \frac{289}{2}
\]
4. **Identify Points with Prime Coordinates**:
- We need to find integer points \((x, y)\) such that:
- \(x + y < 17\)
- Both \(x\) and \(y\) are prime numbers.
5. **List Prime Numbers Less Than 17**:
- The prime numbers less than 17 are: \(2, 3, 5, 7, 11, 13\).
6. **Check Combinations of Prime Coordinates**:
- We will check combinations of these prime numbers to see which pairs \((x, y)\) satisfy \(x + y < 17\):
- \((2, 2)\) → \(2 + 2 = 4\)
- \((2, 3)\) → \(2 + 3 = 5\)
- \((2, 5)\) → \(2 + 5 = 7\)
- \((2, 7)\) → \(2 + 7 = 9\)
- \((2, 11)\) → \(2 + 11 = 13\)
- \((2, 13)\) → \(2 + 13 = 15\)
- \((3, 3)\) → \(3 + 3 = 6\)
- \((3, 5)\) → \(3 + 5 = 8\)
- \((3, 7)\) → \(3 + 7 = 10\)
- \((3, 11)\) → \(3 + 11 = 14\)
- \((3, 13)\) → \(3 + 13 = 16\)
- \((5, 5)\) → \(5 + 5 = 10\)
- \((5, 7)\) → \(5 + 7 = 12\)
- \((5, 11)\) → \(5 + 11 = 16\)
- \((7, 7)\) → \(7 + 7 = 14\)
- \((7, 11)\) → \(7 + 11 = 18\) (not valid)
- \((11, 11)\) → \(11 + 11 = 22\) (not valid)
7. **Count Valid Points**:
- The valid combinations where \(x + y < 17\) are:
- \((2, 2)\)
- \((2, 3)\)
- \((2, 5)\)
- \((2, 7)\)
- \((2, 11)\)
- \((2, 13)\)
- \((3, 3)\)
- \((3, 5)\)
- \((3, 7)\)
- \((3, 11)\)
- \((5, 5)\)
- \((5, 7)\)
- \((7, 7)\)
- Total valid points: 13.
### Final Answer:
The total number of points having coordinates which are prime numbers and lie inside the triangle is **13**.
