Question

A rectangle 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 $A+P?$

$100$

$102$

$104$

$106$

$108$

Answer: B

Concepts:area perimeter Simon’s Favorite Factoring Trick

Difficulty rating: 1540

Solution:

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.

2015 AMC 10A Problem 20

Answer: B

Difficulty rating: 1540

Solution:

Problem 20 in Other Years