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 == | ||
| Line 12: | Line 12: | ||
The points on the line have coordinates | The points on the line have coordinates | ||
<cmath>\left(3+t,17+\frac{88}{15}t\right).</cmath> | <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. | + | 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 | + | == 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 02:22, 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 
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
![\[\frac{88}{15}.\]](http://latex.artofproblemsolving.com/2/8/f/28f0674a136e566eaae5d1be8465410fbefea364.png)
The points on the line have coordinates
![\[\left(3+t,17+\frac{88}{15}t\right).\]](http://latex.artofproblemsolving.com/3/a/2/3a2fd13a96068ad419849ed7b4fe4aafc2ad080b.png)
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 | |
| 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. 
