# 2021 AMC 12B Problems/Problem 10

## 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 $37$ integers is given by $n(n+1)/2$, so $37(37+1)/2=703$.

Therefore, $703-x-y=xy$

Rearranging, $xy+x+y=703$ $(x+1)(y+1)=704$

Looking at the possible divisors of $704 = 2^6*11$, $22$ and $32$ are within the constraints of $0 < x \leq y \leq 37$ so we try those: $(x+1)(y+1) = 22 * 32$ $x+1=22, y+1 = 32$ $x = 21, y = 31$

Therefore, the difference $y-x=31-21=10$, choice E). ~ SoySoy4444

~ pi_is_3.14

~IceMatrix

## See Also

 2021 AMC 12B (Problems • Answer Key • Resources) Preceded byProblem 9 Followed byProblem 11 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions

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