2015 AMC 10A Problem 20

Question

A rectangle with positive integer side lengths in

\mathrm{cm}

has area

A

\mathrm{cm}^2

and perimeter

P

\mathrm{cm}.

Which of the following numbers cannot equal

A+P?

100

102

104

106

108

Factor

A+P+4

as

(x+2)(y+2)

Test the choices after adding

4

Let the side lengths be positive integers

x

and

y

. Then

\begin{aligned} &A+P=xy+2x+2y \\ &=(x+2)(y+2)-4. \end{aligned}

Hence

A+P+4

must factor into two integers both at least

3

.

The answer choices plus

4

are

104,106,108,110,112

. All except

106

have a factorization with both factors at least

3

:

104=4\cdot26,

108=9\cdot12,

110=10\cdot11,

112=7\cdot16.

But

106=2\cdot53

, so it cannot equal

(x+2)(y+2)

.

Thus, B is the correct answer.

Problem 20 in other years