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

Revision as of 10:14, 26 August 2020

The General Statement

Simon's Favorite Factoring Trick (SFF) is often used in a diophantine 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 simple 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 Favorite 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 SFF, 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

Applications

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

Introductory

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?

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

(Source)

Intermediate

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

(Source)

Olympiad

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)

See More

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 ==The General Statement==
+Simon's Favorite Factoring Trick (SFF) is often used in a diophantine 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 simple example would be: <cmath>xy+66x-88y=23333</cmath>where <math>23333</math> is the constant term, <math>xy</math> is the product of the variables, <math>66x</math> and <math>-88y</math> are the variables in linear terms.
+Let's put it in general terms. We have an equation <math>xy+jx+ky=a</math>, where <math>j</math>, <math>k</math>, and <math>a</math> are integral constants. According to Simon's Favorite Factoring Trick, this equation can be transformed into: <cmath>(x+k)(y+j)=a+jk</cmath>
+Using the previous example, <math>xy+66x-88y=23333</math> is the same as: <cmath>(x-88)(y+66)=(23333)(-88)(-66)</cmath>
+If this is confusing or you would like to know the thought process behind SFF, 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
 == Applications ==

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