To determine how many points \((x, y)\) where both \(x\) and \(y\) are positive integers lie below the hyperbola \(xy = 16\), we need to find the integer pairs \((x, y)\) such that \(xy < 16\).
### Step-by-Step Solution:
1. **Consider values of \(x\) and find corresponding \(y\) values**:
For each \(x\), \(y\) must satisfy \(1 \leq y < \frac{16}{x}\).
2. **Calculate pairs for each \(x\)**:
- \(x = 1\):
\[
xy < 16 \implies y < \frac{16}{1} \implies y < 16 \implies y = 1, 2, 3, \ldots, 15 \quad (\text{15 values})
\]
- \(x= 2\):
\[
xy < 16 \implies y < \frac{16}{2} \implies y < 8 \implies y = 1, 2, 3, \ldots, 7 \quad (\text{7 values})
\]
- \(x = 3\):
\[
xy < 16 \implies y < \frac{16}{3} \implies y < 5.33 \implies y = 1, 2, 3, 4, 5 \quad (\text{5 values})
\]
- \(x = 4\):
\[
xy < 16 \implies y < \frac{16}{4} \implies y < 4 \implies y = 1, 2, 3 \quad (\text{3 values})
\]
- \(x = 5\):
\[
xy < 16 \implies y < \frac{16}{5} \implies y < 3.2 \implies y = 1, 2, 3 \quad (\text{3 values})
\]
- \(x = 6\):
\[
xy < 16 \implies y < \frac{16}{6} \implies y < 2.67 \implies y = 1, 2 \quad (\text{2 values})
\]
- \(x = 7\):
\[
xy < 16 \implies y < \frac{16}{7} \implies y < 2.29 \implies y = 1, 2 \quad (\text{2 values})
\]
- \(x = 8\):
\[
xy < 16 \implies y < \frac{16}{8} \implies y < 2 \implies y = 1 \quad (\text{1 value})
\]
- \(x = 9\) and higher:
\[
xy < 16 \implies y < \frac{16}{x} \implies y < \frac{16}{x} \implies y = 1 \quad (\text{1 value if } x \leq 15 \text{ else no values})
\]
3. **Count the total number of pairs**:
Summing all the valid \(y\) values for each \(x\):
\[
15 + 7 + 5 + 3 + 3 + 2 + 2 + 1 + 1 = 39
\]
Therefore, there are \( \boxed{39} \) points of the form \((x, y)\) where both coordinates are positive integers and lie below the hyperbola \(xy = 16\).
