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

Revision as of 18:20, 28 January 2016

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: ${xy}+{xk}+{yj}+{jk}=(x+j)(y+k)$ . Two special common cases are: $xy + x + y + 1 = (x+1)(y+1)$ and $xy - x - y +1 = (x-1)(y-1)$ .

The act of adding ${jk}$ to ${xy}+{xk}+{yj}$ 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 $x$ 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.

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?

$\mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 }$

(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 2 pairs $(x, y)$ of positive integers satisfying:

$\frac 1x +\frac 1y = \frac 1N$

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

@@ Line 24: / Line 24: @@
 ===Olympiad===
-*The integer <math>N</math> is positive. There are exactly 2005 ordered pairs <math>(x, y)</math> of positive integers satisfying:
+*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>

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