AMC12 2021 12B, Problem 10

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?

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 $y-x=31-21=\fbox{10}$. ~ SoySoy4444

Answer: 10