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

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− | *The integer <math>N</math> is positive. There are exactly | + | *The integer <math>N</math> is positive. There are exactly 2 pairs <math>(x, y)</math> of positive integers satisfying: |

<cmath>\frac 1x +\frac 1y = \frac 1N</cmath> | <cmath>\frac 1x +\frac 1y = \frac 1N</cmath> |

## Revision as of 19:20, 28 January 2016

## Contents

## About

**Dr. Simon's Favorite Factoring Trick** (abbreviated **SFFT**) is a special factorization first popularized by AoPS user Simon Rubinstein-Salzedo.

## The General Statement

The general statement of SFFT is: . Two special common cases are: and .

The act of adding to in order to be able to factor it could be called "completing the rectangle" in analogy to the more familiar "completing the square."

## 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

- The integer is positive. There are exactly 2 pairs of positive integers satisfying:

Prove that is a perfect square. (British Mathematical Olympiad Round 2, 2005)