Difference between revisions of "1989 AHSME Problems/Problem 16"
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== Solution == | == Solution == | ||
− | + | The difference in the <math>y</math>-coordinates is <math>281 - 17 = 264</math>, and the difference in the <math>x</math>-coordinates is <math>48 - 3 = 45</math>. | |
+ | The gcd of 264 and 45 is 3, so the line segment joining <math>(3,17)</math> and <math>(48,281)</math> has slope | ||
+ | <cmath>\frac{88}{15}.</cmath> | ||
+ | The points on the line have coordinates | ||
+ | <cmath>\left(3+t,17+\frac{88}{15}t\right).</cmath> | ||
+ | If <math>t</math> is an integer, the <math>y</math>-coordinate of this point is an integer if and only if <math>t</math> is a multiple of 15. The points where <math>t</math> is a multiple of 15 on the segment <math>3\leq x\leq 48</math> are <math>3</math>, <math>3+15</math>, <math>3+30</math>, and <math>3+45</math>. There are 4 lattice points on this line. | ||
== See also == | == See also == |
Revision as of 08:51, 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.
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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