Algebra

I'm getting an answer of 42.  I will post the solution later.

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:
\[
\boxed{44}
\]

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 \(xy = 16\).