2005 AMC 12A Problem 12

Attempt Problem 12 of the 2005 AMC 12A below, then check your answer against the professionally curated solution 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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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

Reduce the slope

\dfrac{1000 - 1}{100 - 1}

to lowest terms

Points on the line take the form

(1 + 11t,\ 1 + 111t);

count integer

t

strictly between the endpoints

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.

Problem 12 in other years