Question

How many points of the form (x,y), where both coordinates are positive integers, lie below the graph of the hyperbola xy = 16?

Answer 1

To find the number of points of the form \( (x, y) \) where both \( x \) and \( y \) are positive integers and lie below the graph of the hyperbola defined by the equation \( xy = 16 \), we start by considering the inequality:

\[
xy < 16
\]

Rearranging this, we get:

\[
y < \frac{16}{x}
\]

Given that \( x \) is a positive integer, we can enumerate values of \( x \) and determine the corresponding integer values of \( y \) based on the above inequality.

Let's compute this for each integer value of \( x \):

1. **For \( x = 1 \)**:
\[
y < \frac{16}{1} = 16 \implies y \in \{1, 2, 3, \ldots, 15\}
\]
So there are **15** valid \( y \)-values.

2. **For \( x = 2 \)**:
\[
y < \frac{16}{2} = 8 \implies y \in \{1, 2, \ldots, 7\}
\]
So there are **7** valid \( y \)-values.

3. **For \( x = 3 \)**:
\[
y < \frac{16}{3} \approx 5.33 \implies y \in \{1, 2, 3, 4, 5\}
\]
So there are **5** valid \( y \)-values.

4. **For \( x = 4 \)**:
\[
y < \frac{16}{4} = 4 \implies y \in \{1, 2, 3\}
\]
So there are **3** valid \( y \)-values.

5. **For \( x = 5 \)**:
\[
y < \frac{16}{5} = 3.2 \implies y \in \{1, 2, 3\}
\]
So there are **3** valid \( y \)-values.

6. **For \( x = 6 \)**:
\[
y < \frac{16}{6} \approx 2.67 \implies y \in \{1, 2\}
\]
So there are **2** valid \( y \)-values.

7. **For \( x = 7 \)**:
\[
y < \frac{16}{7} \approx 2.29 \implies y \in \{1, 2\}
\]
So there are **2** valid \( y \)-values.

8. **For \( x = 8 \)**:
\[
y < \frac{16}{8} = 2 \implies y \in \{1\}
\]
So there is **1** valid \( y \)-value.

9. **For \( x = 9 \)** and higher:
\[
y < \frac{16}{9} \approx 1.78 \implies y \in \{1\}
\]
So there is **1** valid \( y \)-value for each value of \( x \) from \( 9 \) to \( 15 \).

10. **For \( x = 10 \)**:
\[
y < \frac{16}{10} = 1.6 \implies y \in \{1\}
\]
So there is **1** valid \( y \)-value.

11. **For \( x = 11 \)** to **\( x = 15 \)**:
Each yields **1** valid \( y \)-value.

Now we can summarize the valid \( y \)-values for corresponding \( x \):

- \( x = 1 \): 15 points
- \( x = 2 \): 7 points
- \( x = 3 \): 5 points
- \( x = 4 \): 3 points
- \( x = 5 \): 3 points
- \( x = 6 \): 2 points
- \( x = 7 \): 2 points
- \( x = 8 \): 1 point
- \( x = 9 \): 1 point
- \( x = 10 \): 1 point
- \( x = 11 \): 1 point
- \( x = 12 \): 1 point
- \( x = 13 \): 1 point
- \( x = 14 \): 1 point
- \( x = 15 \): 1 point

Now we compute the total:

\[
15 + 7 + 5 + 3 + 3 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 39
\]

Thus, the number of points of the form \( (x, y) \) where both coordinates are positive integers and lie below the graph of \( xy = 16 \) is

\[
\boxed{39}
\]