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\implies \boxed{\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.
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==  Solution 2  ==
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The slope of the line is\[
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\frac{1000-1}{100-1}=\frac{111}{11},
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\]so all points on the line have the form <math>(1+11t, 1+111t)</math> for some value of <math>t</math> (the rise is 111 and the run is 11). Such a point has integer coordinates if and only if <math>t</math> is an integer, and the point is strictly between <math>A</math> and <math>B</math> if and only if <math>0<t<9</math>. Thus, there are <math>\boxed{8}</math> points with the required property.
  
 
== See also ==
 
== See also ==

Revision as of 11:41, 28 May 2021

Problem

A line passes through $A\ (1,1)$ and $B\ (100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$?

$(\mathrm {A}) \ 0 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 3 \qquad (\mathrm {D}) \ 8 \qquad (\mathrm {E})\ 9$

Solution

For convenience’s sake, we can transform $A$ to the origin and $B$ to $(99,999)$ (this does not change the problem). The line $AB$ has the equation $y = \frac{999}{99}x = \frac{111}{11}x$. The coordinates are integers if $11|x$, so the values of $x$ are $11, 22 \ldots 88$, with a total of $8\implies \boxed{\mathrm{(D)}}$ 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 $(1+11t, 1+111t)$ for some value of $t$ (the rise is 111 and the run is 11). Such a point has integer coordinates if and only if $t$ is an integer, and the point is strictly between $A$ and $B$ if and only if $0<t<9$. Thus, there are $\boxed{8}$ points with the required property.

See also

2005 AMC 12A (ProblemsAnswer KeyResources)
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

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