Difference between revisions of "Simon's Favorite Factoring Trick"
(clean up) |
|||
Line 1: | Line 1: | ||
{{WotWAnnounce|week=August 22-August 28}} | {{WotWAnnounce|week=August 22-August 28}} | ||
− | '''Simon's Favorite Factoring Trick''' (abbreviated '''SFFT''') is a special factorization first popularized by [[AoPS]] user [[user:ComplexZeta | Simon Rubinstein-Salzedo]]. <url>viewtopic.php?highlight=factoring&t=8215 This</url> appears to be the thread where Simon's favorite factoring trick was first introduced. | + | '''Simon's Favorite Factoring Trick''' (abbreviated '''SFFT''') is a special factorization first popularized by [[AoPS]] user [[user:ComplexZeta | Simon Rubinstein-Salzedo]]. <url>viewtopic.php?highlight=factoring&t=8215 This</url> appears to be the thread where Simon's favorite factoring trick was first introduced. The general statement of SFFT is: <math>{xy}+{xk}+{yj}+{jk}=(x+j)(y+k)</math>. Two special common cases are: <math>xy + x + y + 1 = (x+1)(y+1)</math> and <math>xy - x - y +1 = (x-1)(y-1)</math>. |
− | |||
− | |||
− | The general statement of SFFT is: <math>{xy}+{xk}+{yj}+{jk}=(x+j)(y+k)</math>. Two special cases | ||
== Applications == | == Applications == | ||
− | This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually <math> | + | This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually <math>x</math> and <math>y</math> are variables and <math>j,k</math> are known constants. Also, it is typically necessary to add the <math>jk</math> term to both sides to perform the factorization. |
== Problems == | == Problems == | ||
===Introductory=== | ===Introductory=== | ||
− | * | + | *Two different [[prime number]]s between <math>4</math> and <math>18</math> are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? |
− | Two different [[prime number]]s between <math>4</math> and <math>18</math> are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? | ||
<math> \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 } </math> | <math> \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 } </math> |
Revision as of 22:26, 27 August 2008
This is an AoPSWiki Word of the Week for August 22-August 28 |
Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization first popularized by AoPS user Simon Rubinstein-Salzedo. <url>viewtopic.php?highlight=factoring&t=8215 This</url> appears to be the thread where Simon's favorite factoring trick was first introduced. The general statement of SFFT is: . Two special common cases are: and .
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)
Olympiad
This problem has not been edited in. If you know this problem, please help us out by adding it.