To determine the number of points of the form \((x,y)\), where both coordinates are positive integers, that lie below the graph of the hyperbola \(xy = 16\), we need to find the integer pairs \((x, y)\) such that \(xy < 16\).
### Step-by-Step Solution:
1. **Identify the Range for \(x\)**:
- For each \(x\), we need \(y\) to be an integer such that \(xy < 16\).
- Since \(x\) must be a positive integer, consider possible values of \(x\) starting from 1 up to the point where \(x \cdot 1 = 16\), so \(x\) ranges from 1 to 15.
2. **Count \(y\) Values for Each \(x\)**:
- For each \(x\), find the largest integer \(y\) such that \(y < \frac{16}{x}\).
Here’s how this works for each \(x\) from 1 to 15:
- \(x = 1\): \(xy < 16 \implies y < \frac{16}{1} = 16\). So, \(y\) can be 1 to 15 (15 values).
- \(x = 2\): \(xy < 16 \implies y < \frac{16}{2} = 8\). So, \(y\) can be 1 to 7 (7 values).
- \(x = 3\): \(xy < 16 \implies y < \frac{16}{3} \approx 5.33\). So, \(y\) can be 1 to 5 (5 values).
- \(x = 4\): \(xy < 16 \implies y < \frac{16}{4} = 4\). So, \(y\) can be 1 to 3 (3 values).
- \(x = 5\): \(xy < 16 \implies y < \frac{16}{5} = 3.2\). So, \(y\) can be 1 to 3 (3 values).
- \(x = 6\): \(xy < 16 \implies y < \frac{16}{6} \approx 2.67\). So, \(y\) can be 1 to 2 (2 values).
- \(x = 7\): \(xy < 16 \implies y < \frac{16}{7} \approx 2.29\). So, \(y\) can be 1 to 2 (2 values).
- \(x = 8\): \(xy < 16 \implies y < \frac{16}{8} = 2\). So, \(y\) can be 1 (1 value).
- \(x = 9\) to \(x = 15\): For these values, \(y < \frac{16}{x}\) will always be less than 2, so \(y\) can only be 1 (1 value each).
3. **Summarize the Counts**:
\[
\begin{aligned}
&15 \text{ values for } x = 1, \\
&7 \text{ values for } x = 2, \\
&5 \text{ values for } x = 3, \\
&3 \text{ values for } x = 4, \\
&3 \text{ values for } x = 5, \\
&2 \text{ values for } x = 6, \\
&2 \text{ values for } x = 7, \\
&1 \text{ value for } x = 8, \\
&1 \text{ value each for } x = 9 \text{ to } 15.
\end{aligned}
\]
Adding these up:
\[
15 + 7 + 5 + 3 + 3 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 44
\]
Therefore, the number of points \((x, y)\), where both coordinates are positive integers, that lie below the graph of the hyperbola \(xy = 16\) is:
\[
\boxed{44}
\]
