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 \( x \):
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For \( x = 1 \): \[ y < \frac{16}{1} = 16 \implies y \in {1, 2, 3, \ldots, 15} \] So there are 15 valid \( y \)-values.
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For \( x = 2 \): \[ y < \frac{16}{2} = 8 \implies y \in {1, 2, \ldots, 7} \] So there are 7 valid \( y \)-values.
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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.
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For \( x = 4 \): \[ y < \frac{16}{4} = 4 \implies y \in {1, 2, 3} \] So there are 3 valid \( y \)-values.
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For \( x = 5 \): \[ y < \frac{16}{5} = 3.2 \implies y \in {1, 2, 3} \] So there are 3 valid \( y \)-values.
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For \( x = 6 \): \[ y < \frac{16}{6} \approx 2.67 \implies y \in {1, 2} \] So there are 2 valid \( y \)-values.
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For \( x = 7 \): \[ y < \frac{16}{7} \approx 2.29 \implies y \in {1, 2} \] So there are 2 valid \( y \)-values.
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For \( x = 8 \): \[ y < \frac{16}{8} = 2 \implies y \in {1} \] So there is 1 valid \( y \)-value.
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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 \( 15 \).
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For \( x = 10 \): \[ y < \frac{16}{10} = 1.6 \implies y \in {1} \] So there is 1 valid \( y \)-value.
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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
\[ \boxed{39} \]
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