Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1989 AHSME Problems/Problem 16"

m
(→‎Solution)
 
(9 intermediate revisions by 5 users not shown)
Line 1: Line 1:
 
== Problem ==
 
== 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 (3,17) and (48,281)? (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> \textrm{(A)}\ 2\qquad\textrm{(B)}\ 4\qquad\textrm{(C)}\ 6\qquad\textrm{(D)}\ 16\qquad\textrm{(E)}\ 46 </math>
 
  
 +
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 46 </math>
  
 
== Solution ==
 
== Solution ==
  
−
Since the endpoints are (3,17) and (48,281), the line that passes through these 2 points has slope <math>m=\frac{281-17}{48-3}=\frac{264}{45}=\frac{88}{15}</math>. The equation of the line passing through these points can then be given by <math>y=17+\frac{88}{15}(x-3)</math>. Since <math>\frac{88}{15}</math> is reduced to lowest terms, in order for <math>y</math> to be integral we must have that <math>15|x-3</math>. Hence <math>x</math> is 3 more than a multiple of 15. Note that <math>x=3</math> corresponds to the endpoint <math>(3,17)</math>. Then we have <math>x=18</math>, <math>x=33</math>, and <math>x=48</math> where <math>x=48</math> corresponds to the endpoint <math>(48,281)</math>. Hence there are 4 in all.
+
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. Hence the answer is <math>\boxed{B}</math>.
  
−
== See also ==
+
== See Also ==
 
{{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}}

Latest revision as of 03:22, 11 July 2020

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 $(3,17)$ and $(48,281)$? (Include both endpoints of the segment in your count.)

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 46$

Solution

The difference in the $y$-coordinates is $281 - 17 = 264$, and the difference in the $x$-coordinates is $48 - 3 = 45$. The gcd of 264 and 45 is 3, so the line segment joining $(3,17)$ and $(48,281)$ has slope \[\frac{88}{15}.\] The points on the line have coordinates \[\left(3+t,17+\frac{88}{15}t\right).\] If $t$ is an integer, the $y$-coordinate of this point is an integer if and only if $t$ is a multiple of 15. The points where $t$ is a multiple of 15 on the segment $3\leq x\leq 48$ are $3$, $3+15$, $3+30$, and $3+45$. There are 4 lattice points on this line. Hence the answer is $\boxed{B}$.

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1989_AHSME_Problems/Problem_16&oldid=128045"