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

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This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually <math>\displaystyle {x}</math> and <math>\displaystyle {y}</math> are variables and <math>\displaystyle j,k</math> are known constants. Also it is typically necessary to add the <math>\displaystyle {j}{k}</math> term to both sides to perform the factorization. | This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually <math>\displaystyle {x}</math> and <math>\displaystyle {y}</math> are variables and <math>\displaystyle j,k</math> are known constants. Also it is typically necessary to add the <math>\displaystyle {j}{k}</math> term to both sides to perform the factorization. | ||

− | == | + | == Problems == |

+ | ===Introductory=== | ||

+ | * | ||

+ | Two different [[prime number]]s between <math>4</math> and <math>18</math> are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? | ||

− | + | <math> \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 } </math> | |

− | + | ([[2000 AMC 12/Problem 6|Source]]) | |

+ | ===Intermediate=== | ||

+ | *<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]]) | ||

+ | ===Olympiad=== | ||

+ | {{problem}} | ||

== See Also == | == See Also == |

## Revision as of 17:10, 10 November 2007

## Contents

## Introduction

**Simon's Favorite Factoring Trick** (abbreviated SFFT) is a special factorization first popularized by AoPS user Simon Rubinstein-Salzedo. <url>viewtopic.php?highlight=factoring&t=8215 This</url> appears to be the thread where Simon's favorite factoring trick was first introduced.

## Statement of the factorization

The general statement of SFFT is: . Two special cases appear most commonly: and .

## Applications

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

## Problems

### Introductory

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

(Source)

### Intermediate

- are integers such that . Find .

(Source)

### Olympiad

*This problem has not been edited in. If you know this problem, please help us out by adding it.*