2005 AMC 12A Problem 12

Below is the professionally curated solution for Problem 12 of the 2005 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 12A solutions, or check the answer key.

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Concepts:lattice point greatest common divisor slope

Difficulty rating: 1660

12.

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 $B?$

$0$

$2$

$3$

$8$

$9$

Solution:

The slope is $\dfrac{1000 - 1}{100 - 1} = \dfrac{999}{99} = \dfrac{111}{11}.$

So every point on the line has the form $(1 + 11t,\ 1 + 111t),$ which is a lattice point exactly when $t$ is an integer. The point is strictly between $A$ and $B$ when $0 \lt t \lt 9.$

There are $8$ such integers $t,$ giving $8$ lattice points.

Thus, the correct answer is D.

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