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 $xy = 16$.