Can someone please help?

To count the points (x, y) with positive integer coordinates that lie below the hyperbola xy = 16, we can consider the following:

 

The hyperbola xy = 16: This is a downward-sloping curve that passes through the points (1, 16), (2, 8), (4, 4), (8, 2), and (16, 1). Any point below this curve will have a product of coordinates less than 16.

 

Restricting to positive integers: Since we only want points with positive integer coordinates, we can focus on the region in the first quadrant where both x and y are positive.

 

Counting the points: We can systematically count the points in this region that satisfy xy < 16:

 

For x = 1, we can have y = 1, 2, ..., 15 (15 points).

 

For x = 2, we can have y = 1, 2, ..., 7 (7 points).

 

For x = 3, we can have y = 1, 2, ..., 5 (5 points).

 

For x = 4, we can have y = 1, 2, 3 (3 points).

 

For x = 5, we can have y = 1, 2, 3 (3 points).

 

For x = 6, we can have y = 1, 2 (2 points).

 

For x = 7, we can have y = 1, 2 (2 points).

 

For x = 8, we can have y = 1 (1 point).

 

Summing these up, we get 15 + 7 + 5 + 3 + 3 + 2 + 2 + 1 = 43 points.

 

Therefore, there are 43 points of the form (x, y), where both coordinates are positive integers, that lie below the graph of the hyperbola xy = 16.

To find how many points of the form \((x, y)\) lie below the graph of the hyperbola defined by the equation \(xy = 16\), we start by expressing \(y\) in terms of \(x\):

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

We want to find integer points \((x, y)\) where both \(x\) and \(y\) are positive integers, and \(y < \frac{16}{x}\). Hence, we can reformulate our problem to determine the constraints on \(x\):

1. \(y\) must be a positive integer:

To ensure \(y\) is an integer, \(x\) must be a divisor of \(16\).

2. The divisors of \(16\) can be calculated:
The positive divisors of \(16\) are \(1, 2, 4, 8, 16\).

Now we will determine \(y\) for each divisor of \(16\) and check how many pairs \((x, y)\) we can find such that \(y\) remains below \(\frac{16}{x}\):

\[
\begin{align*}
x = 1 & \Rightarrow y = \frac{16}{1} = 16 \quad (\text{valid since } y < 16)\\
x = 2 & \Rightarrow y = \frac{16}{2} = 8 \quad (\text{valid since } y < 8)\\
x = 4 & \Rightarrow y = \frac{16}{4} = 4 \quad (\text{valid since } y < 4)\\
x = 8 & \Rightarrow y = \frac{16}{8} = 2 \quad (\text{valid since } y < 2)\\
x = 16 & \Rightarrow y = \frac{16}{16} = 1 \quad (\text{valid since } y < 1)\\
\end{align*}
\]

Next, we need to determine how many integer values of \(y\) are valid (i.e., \(y\) must be positive and lie below the computed value):

- For \(x = 1\), \(y < 16\) gives valid values: \(1, 2, \ldots, 15\) (15 values).

- For \(x = 2\), \(y < 8\) gives valid values: \(1, 2, \ldots, 7\) (7 values).

- For \(x = 4\), \(y < 4\) gives valid values: \(1, 2, 3\) (3 values).

- For \(x = 8\), \(y < 2\) gives valid values: \(1\) (1 value).

- For \(x = 16\), \(y < 1\) yields no positive integers.

Now we add up the number of valid \(y\) values:

\[
15 + 7 + 3 + 1 + 0 = 26
\]

Thus, the total number of points \((x, y)\) where both \(x\) and \(y\) are positive integers that lie below the hyperbola defined by \(xy = 16\) is:

\[
\boxed{26}
\]