SFFT or Simon's Favorite Factoring Trick

Algebra Progress

Completed0/18

0 completed0 in progress

Simon's Favorite Factoring Trick's formula and its applicatinos.

Author: Jaden Park

Formula

SFFT, or Simon's favorite factoring trick, is used in Diophantine equations you need to factor.

The general formula is for an equation $xy + px + qy = r,$ where $p, q,$ and $r$ are integer constants, and the coefficeint of $xy$ must be 1 (if it's not $1,$ divide the coefficient), the equation can be rewritten as:

(x + q)(y + p) = r + pq.

Let's do a small example.

Consider the equation $xy + 10x - 45y = 1000.$

Using SFFT, we get

(x - 45)(y + 10) = 1000 + (-45)(10).

Here are some practice problems.

Practice Problem

(2021 CEMC Galois Q4b)

Determine all possible ordered pairs of possitive integers $(x, y)$ that are solutions to the equation

\frac{4}{x} + \frac{5}{y} = 1.

Practice Problem

(1987 AIME Q5)

Find $3x^{2}y^{2}$ if $x$ and $y$ are integers such that $y^{2} + 3x^{2}y^{2} = 30x^{2} + 517.$

Practice Problem

(2005 British Mathematical Olympiad Round 2 Q1)

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

\frac{1}{x} + \frac{1}{y} = \frac{1}{N}

Prove that $N$ is a perfect square.

Final Notes and Tips

Not that common but I would keep it in the back of my head incase the question has a diophantine equation.

Differences and Sums of Powers

Factor Theorem

Algebra Progress

Polynomials

Sequence and Series

Miscellaneous

Formula

Practice Problem

Practice Problem

Practice Problem

Final Notes and Tips

Formula

Practice Problem

Practice Problem

Practice Problem

Final Notes and Tips