Difference between revisions of "1989 AHSME Problems/Problem 16"
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== 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> \textrm{(A)}\ 2\qquad\textrm{(B)}\ 4\qquad\textrm{(C)}\ 6\qquad\textrm{(D)}\ 16\qquad\textrm{(E)}\ 46 </math> | ||
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== Solution == | == Solution == | ||
Revision as of 17:50, 14 January 2016
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
Since the endpoints are (3,17) and (48,281), the line that passes through these 2 points has slope 
. The equation of the line passing through these points can then be given by 
. Since 
is reduced to lowest terms, in order for 
to be integral we must have that 
. Hence 
is 3 more than a multiple of 15. Note that 
corresponds to the endpoint . Then we have 
, 
, and 
where corresponds to the endpoint . Hence there are 4 in all.
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. 
