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

Revision as of 00:05, 10 November 2015

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."

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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. BID DJD<DlkDJDHDdcc

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 2005 ordered 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)

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