2005 AMC 12A Problem 12

Problem 12 of 25IntermediateAlgebra Geometry Number Theory

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?

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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.

Tagged: lattice point · greatest common divisor · slope