2021 AMC 12B Problems/Problem 10

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Problem

Two distinct numbers are selected from the set $\{1,2,3,4,\dots,36,37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers?

$\textbf{(A) }5 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8\qquad \textbf{(D) }9 \qquad \textbf{(E) }10$

Solution

The sum of the first $n$ integers is given by $\frac{n(n+1)}{2}$, so $\frac{37(37+1)}{2}=703$.

Therefore, $703-x-y=xy$

Rearranging, $xy+x+y=703$. We can factor this equation by SFFT to get

$(x+1)(y+1)=704$

Looking at the possible divisors of $704 = 2^6\cdot11$, $22$ and $32$ are within the constraints of $0 < x \leq y \leq 37$ so we try those:

$(x+1)(y+1) = 22\cdot32$

$x+1=22, y+1 = 32$

$x = 21, y = 31$

Therefore, the difference $y-x=31-21=\boxed{\textbf{(E) }10}$.

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