To solve the problem, we need to find the number of points \( C(x, y) \) on the circle defined by the equation \( x^2 + y^2 = 16 \) such that the area of triangle \( ABC \) is a positive integer. The points \( A \) and \( B \) are given as \( A(-4, 0) \) and \( B(4, 0) \).
### Step-by-Step Solution:
1. **Identify the Circle's Properties**:
The equation \( x^2 + y^2 = 16 \) represents a circle centered at the origin \( (0, 0) \) with a radius \( r = 4 \).
2. **Determine the Area of Triangle \( ABC \)**:
The area \( A \) of triangle \( ABC \) can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base \( AB \) is the distance between points \( A \) and \( B \):
\[
AB = |x_B - x_A| = |4 - (-4)| = 8
\]
The height is the perpendicular distance from point \( C(x, y) \) to line \( AB \) (which lies on the x-axis). Thus, the height is simply the y-coordinate of point \( C \), which is \( y \).
Therefore, the area of triangle \( ABC \) is:
\[
\text{Area} = \frac{1}{2} \times 8 \times |y| = 4 |y|
\]
3. **Set the Area to be a Positive Integer**:
For the area to be a positive integer, we can write:
\[
4 |y| = k \quad \text{(where \( k \) is a positive integer)}
\]
This implies:
\[
|y| = \frac{k}{4}
\]
4. **Find Possible Values of \( k \)**:
Since \( C(x, y) \) lies on the circle \( x^2 + y^2 = 16 \), we have:
\[
y^2 = 16 - x^2
\]
Substituting \( |y| = \frac{k}{4} \):
\[
\left(\frac{k}{4}\right)^2 = 16 - x^2
\]
Simplifying gives:
\[
\frac{k^2}{16} + x^2 = 16
\]
\[
x^2 = 16 - \frac{k^2}{16}
\]
\[
x^2 = \frac{256 - k^2}{16}
\]
5. **Determine the Range for \( k \)**:
For \( x^2 \) to be non-negative:
\[
256 - k^2 \geq 0 \implies k^2 \leq 256 \implies |k| \leq 16
\]
Since \( k \) is a positive integer, \( k \) can take values from \( 1 \) to \( 16 \).
6. **Count Valid Integer Values of \( k \)**:
The possible values of \( k \) are \( 1, 2, 3, \ldots, 16 \). This gives us a total of \( 16 \) possible values.
7. **Determine Corresponding \( y \) Values**:
For each \( k \), \( |y| = \frac{k}{4} \) gives two possible \( y \) values (positive and negative) except for \( k = 0 \), which is not considered since we want positive area. Thus, for each \( k \) from \( 1 \) to \( 16 \), we have two corresponding \( y \) values.
8. **Final Count of Points**:
Since each \( k \) gives us two points \( (x, y) \) for each \( k \) (one for \( y \) and one for \( -y \)), the total number of points \( C \) is:
\[
2 \times 16 = 32
\]
### Conclusion:
The number of points \( C(x, y) \) on the circle such that the area of triangle \( ABC \) is a positive integer is **32**.
