Difference between revisions of "Simon's Favorite Factoring Trick"

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*<math>m, n</math> are integers such that <math>m^2 + 3m^2n^2 = 30n^2 + 517</math>. Find <math>3m^2n^2</math>.
*<math>m, n</math> are integers such that <math>m^2 + 3m^2n^2 = 30n^2 + 517</math>. Find <math>3m^2n^2</math>.
([[1987 AIME Problems/Problem 5|Source]])
([[1987 AIME Problems/Problem 5|Source]]) NERD

Revision as of 19:51, 26 March 2021

The General Statement

Simon's Favorite Factoring Trick (SFFT) is often used in a Diophantine(Positive) equation where factoring is needed. The most common form it appears is when there is a constant on one side of the equation and a product of variables with each of those variables in a linear term on the other side. A extortive example would be: \[xy+66x-88y=23333\]where $23333$ is the constant term, $xy$ is the product of the variables, $66x$ and $-88y$ are the variables in linear terms.

Let's put it in general terms. We have an equation $xy+jx+ky=a$, where $j$, $k$, and $a$ are integral constants. According to Simon's Favourite Factoring Trick, this equation can be transformed into: \[(x+k)(y+j)=a+jk\] Using the previous example, $xy+66x-88y=23333$ is the same as: \[(x-88)(y+66)=(23333)+(-88)(66)\]

If this is confusing or you would like to know the thought process behind SFFT, see this eight-minute video by Richard Rusczyk from AoPS: https://www.youtube.com/watch?v=0nN3H7w2LnI. For the thought process, start from https://youtu.be/0nN3H7w2LnI?t=366


This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually $x$ and $y$ are variables and $j,k$ are known constants. Also, it is typically necessary to add the $jk$ term to both sides to perform the factorization.

Fun Practice Problems


  • Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?

$\mathrm{(A) \ 22 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }$



  • $m, n$ are integers such that $m^2 + 3m^2n^2 = 30n^2 + 517$. Find $3m^2n^2$.

(Source) NERD


  • The integer $N$ is positive. There are exactly 2005 ordered pairs $(x, y)$ of positive integers satisfying:

\[\frac 1x +\frac 1y = \frac 1N\]

Prove that $N$ is a perfect square.

Source: (British Mathematical Olympiad Round 3, 2005)

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