How many lattice points (points with integer coordinates) are on the line segment whose endpoints are $(3,17)$ and $(82,150)?$ (Include both endpoints in your count.)

ChiIIBill Jul 23, 2024

#1**+1 **

First, let's figure out the slope of the two points.

Since the slope is \(\frac{y_2-y_1}{x_2-x_1}\), the slope for (3,17) and (82,150) is

\(\frac{150-17}{82-3} = \frac{133}{79}\)

However, notice that between the two points, there are no lattice points.

This is because following the slope, after (3, 17), the next latice point is

\((3+79, 17+133) = (82, 150)\)

Thus, the two endpoints are the only latice points, there are 2.

Thanks! :)

NotThatSmart Jul 23, 2024

#1**+1 **

Best Answer

First, let's figure out the slope of the two points.

Since the slope is \(\frac{y_2-y_1}{x_2-x_1}\), the slope for (3,17) and (82,150) is

\(\frac{150-17}{82-3} = \frac{133}{79}\)

However, notice that between the two points, there are no lattice points.

This is because following the slope, after (3, 17), the next latice point is

\((3+79, 17+133) = (82, 150)\)

Thus, the two endpoints are the only latice points, there are 2.

Thanks! :)

NotThatSmart Jul 23, 2024