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 | + | [[Category: Introductory Number Theory Problem]] |
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 20:55, 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
.
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.