Difference between revisions of "2005 AMC 12A Problems/Problem 12"
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== Solution == | == Solution == | ||
| − | For convenience’s sake, we can transform <math>A</math> to the origin and <math>B</math> to <math>(99,999)</math> (this does not change the problem). The line <math>AB</math> has the [[equation]] <math>y = \frac{999}{99}x = \frac{111}{11}x</math>. The coordinates are integers if <math>11|x</math>, so the values of <math>x</math> are <math>11, 22 \ldots 88</math>, with a total of <math>8\ \mathrm{(D)}</math> coordinates. | + | For convenience’s sake, we can transform <math>A</math> to the origin and <math>B</math> to <math>(99,999)</math> (this does not change the problem). The line <math>AB</math> has the [[equation]] <math>y = \frac{999}{99}x = \frac{111}{11}x</math>. The coordinates are integers if <math>11|x</math>, so the values of <math>x</math> are <math>11, 22 \ldots 88</math>, with a total of <math>8\implies \boxed{\mathrm{(D)}}</math> coordinates. |
== See also == | == See also == | ||
Revision as of 13:58, 7 November 2019
Problem
A line passes through 
and 
. How many other points with integer coordinates are on the line and strictly between 
and 
?
Solution
For convenience’s sake, we can transform to the origin and to 
(this does not change the problem). The line 
has the equation 
. The coordinates are integers if 
, so the values of 
are 
, with a total of 
coordinates.
See also
| 2005 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 11 |
Followed by Problem 13 |
| 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 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
