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# Algebra

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

Jul 24, 2024

#1
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I'm getting an answer of 42.  I will post the solution later.

Jul 24, 2024
edited by AnswerscorrectIy  Jul 24, 2024
#3
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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}

$15 + 7 + 5 + 3 + 3 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 44$

Therefore, the number of points $$(x, y)$$, where both coordinates are positive integers, that lie below the graph of the hyperbola $$xy = 16$$ is:
$\boxed{44}$

Jul 24, 2024
#4
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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 $$xy = 16$$.

Jul 24, 2024