1989 AHSME Problems/Problem 16
Problem
A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are and ? (Include both endpoints of the segment in your count.)
Solution
The difference in the -coordinates is , and the difference in the -coordinates is . The gcd of 264 and 45 is 3, so the line segment joining and has slope The points on the line have coordinates If is an integer, the -coordinate of this point is an integer if and only if is a multiple of 15. The points where is a multiple of 15 on the segment are , , , and . There are 4 lattice points on this line. Hence the answer is .
See Also
1989 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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