2005 AMC 12A Problems/Problem 12
Contents
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.
Solution 2
The slope of the line is\[ \frac{1000-1}{100-1}=\frac{111}{11}, \]so all points on the line have the form for some value of (the rise is 111 and the run is 11). Such a point has integer coordinates if and only if is an integer, and the point is strictly between and if and only if . Thus, there are points with the required property. -Paixiao
See also
2005 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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All AMC 12 Problems and Solutions |
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