Difference between revisions of "1989 AHSME Problems/Problem 16"
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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. Hence the answer <math>\textbf{(B)}</math>. | 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. Hence the answer <math>\textbf{(B)}</math>. | ||
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{{AHSME box|year=1989|num-b=15|num-a=17}} | {{AHSME box|year=1989|num-b=15|num-a=17}} | ||
[[Category: Introductory Number Theory Problems]] | [[Category: Introductory Number Theory Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 20:54, 21 January 2019
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 .
seeee allso
1989 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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