Question

Rodrigo has a very large piece of graph paper. First he draws a line segment connecting point $(0, 4)$ to point $(2, 0)$ and colors the $4$ cells whose interiors intersect the segment, as shown below. Next, Rodrigo draws a line segment connecting point $(2000, 3000)$ to point $(5000, 8000).$ Again he colors the cells whose interiors intersect the segment. How many cells will he color this time?

$6000$

$6500$

$7000$

$7500$

$8000$

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Written solution:

For a segment whose endpoint differences are $a$ horizontally and $b$ vertically, the segment crosses $a$ vertical grid lines and $b$ horizontal grid lines, but crossings at lattice points are counted twice. Therefore the number of cells whose interiors are intersected is $a+b-\gcd(a,b).$

Here the endpoint differences are $3000$ and $5000$ , and $\gcd(3000,5000)=1000$ . Thus the number of cells colored is $3000+5000-1000=7000.$

Thus, C is the correct answer.

2024 AMC 8 Problem 23

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Problem 23 in Other Years