Difference between revisions of "1989 AHSME Problems/Problem 16"
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A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are <math>(3,17)</math> and <math>(48,281)</math>? (Include both endpoints of the segment in your count.) | A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are <math>(3,17)</math> and <math>(48,281)</math>? (Include both endpoints of the segment in your count.) | ||
− | <math> \ | + | <math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 46 </math> |
== Solution == | == Solution == |
Revision as of 08:59, 2 April 2018
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 .
See also
1989 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
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