FidgetBoss 4000's 2019 AMC 12B Problems/Problem 1

Problem

If $x$ and $y$ are positive integers such that their product is $64,$ what is the sum of all distinct values for $xy+x+y?$

$\textbf{(A) }391\qquad\textbf{(B) }392\qquad\textbf{(C) }393\qquad\textbf{(D) }394\qquad\textbf{(E) }395\qquad$

Solution 1

Use Simon's Favorite Factoring Trick to deduce $xy+x+y=(x+1)(y+1)-1$ . We know $x$ and $y$ are positive integers and $xy=64$ . So, we have the following possibilities for the pair $(x, y): (1, 64), (2, 32), (4, 16), (8, 8)$ . Note that we do not have to account for flipping $x$ and $y$ because the question asks for only the $\emph{distinct}$ values for $xy+x+y$ . If $(x, y)=(1, 64)$ , then $(x+1)(y+1)-1=2\cdot 65-1=129$ . If $(x, y)=(2, 32)$ , then $3\cdot 33-1=98$ . If $(x, y)=(4, 16)$ , then $5\cdot 17-1=84$ . If $(x, y)=(8, 8)$ , then $9\cdot 9-1=80$ . The sum of these distinct values is $129+98+84+80=227+164=\boxed{391}\textbf{(A)}$ .

2019 AMC 12B (Problems • Answer Key • Resources)
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FidgetBoss 4000's 2019 AMC 12B Problems/Problem 1

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Solution 1

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