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 $A+P$? $\textbf{(A) }100\qquad$ $\textbf{(B) }102\qquad$ $\textbf{(C) }104\qquad$ $\textbf{(D) }106\qquad$ $\textbf{(E) }108$
Solution
Assume the side lengths are $a, b$. Then $A=ab, P=2a+2b$.
$A+P=ab+2a+2b=(a+2)(b+2)-4$.
Since $a,b\ge1$, both factors $a+2$, $b+2$ are at least $3$. For $A+P=102$, $(a+2)(b+2)=106=2\cdot 53$ has no such factorization. The others work: $104=4\cdot26$, $108=3\cdot36$, $110=10\cdot11$, $112=4\cdot28$.
