2008 AMC 12B Problem 16

Question

A rectangular floor measures

a

feet by

b

feet, where

a

and

b

are positive integers with

b \gt a.

An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width

1

foot around the painted rectangle and occupies half the area of the entire floor. How many possibilities are there for the ordered pair

(a, b)?

1

2

3

4

5

Answer: B

Concepts:area Simon’s Favorite Factoring Trick factor

Difficulty rating: 1660

Small hint:

The painted rectangle is

(a - 2) \times (b - 2)

and equals half the floor, so

ab = 2(a - 2)(b - 2)

Big hint:

Rearrange to

(a - 4)(b - 4) = 8

and count factor pairs with

b \gt a \gt 0

Solution:

The painted rectangle measures

(a - 2)

by

(b - 2)

and has half the area of the floor, so

ab = 2(a - 2)(b - 2).

Expanding gives

0 = ab - 4a - 4b + 8,

and adding

8

yields

(a - 4)(b - 4) = 8.

With

b \gt a \gt 0,

the only valid factor pairs of

8

are

(a - 4, b - 4) = (1, 8)

and

(2, 4),

giving

(a, b) = (5, 12)

and

(6, 8).

There are

2

possibilities.

Thus, the correct answer is B.