Difference between revisions of "Simon's Favorite Factoring Trick"
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*<math>m, n</math> are integers such that <math>m^2 + 3m^2n^2 = 30n^2 + 517</math>. Find <math>3m^2n^2</math>. | *<math>m, n</math> are integers such that <math>m^2 + 3m^2n^2 = 30n^2 + 517</math>. Find <math>3m^2n^2</math>. |
Revision as of 22:46, 17 December 2016
Contents
[hide]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)
- The integer
is positive. There are exactly
pairs
of positive integers satisfying:
Prove that is a perfect square. (British Mathematical Olympiad Round 2, 2005)