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# Number Theory

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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.)

Jul 23, 2024

#1
+1790
+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! :)

Jul 23, 2024
edited by NotThatSmart  Jul 23, 2024

#1
+1790
+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
edited by NotThatSmart  Jul 23, 2024