2015 AMC 10A Problems/Problem 20

Problem

A square with positive integer side lengths in $\text{cm}$ has area $A$ $\text{cm}^2$ and perimeter $P$ $\text{cm}$ . Which of the following numbers cannot equal $A+P$ ?

$\textbf{(A) }100\qquad\textbf{(B) }102\qquad\textbf{(C) }104\qquad\textbf{(D) }106\qquad\textbf{(E) }108$

Solution

Let the rectangle's length and width be $a$ and $b$ . Its area is $ab$ and the perimeter is $2(a + b)$ .

Then $A + P = ab + 2a + 2b$ . Factoring, this is $(a + 2)(b + 2) - 4$ .

Looking at the answer choices, only $102$ cannot be written this way, because then either $a$ or $b$ would be $0$ .

So the answer is $\boxed{\textbf{(B) }102}$ .

2015 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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2015 AMC 10A Problems/Problem 20

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