https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Ikitten&feedformat=atom AoPS Wiki - User contributions [en] 2021-04-16T23:46:22Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=Simon%27s_Favorite_Factoring_Trick&diff=88272 Simon's Favorite Factoring Trick 2017-11-14T17:44:43Z <p>Ikitten: /* The General Statement */</p> <hr /> <div><br /> ==About==<br /> '''Dr. Simon's Favorite Factoring Trick''' (abbreviated '''SFFT''') is a special factorization first popularized by [[AoPS]] user [[user:ComplexZeta | Simon Rubinstein-Salzedo]].<br /> <br /> ==The General Statement==<br /> The general statement of SFFT is: &lt;math&gt;{xy}+{xk}+{jy}+{jk}=(x+j)(y+k)&lt;/math&gt;. Two special common cases are: &lt;math&gt;xy + x + y + 1 = (x+1)(y+1)&lt;/math&gt; and &lt;math&gt;xy - x - y +1 = (x-1)(y-1)&lt;/math&gt;.<br /> <br /> The act of adding &lt;math&gt;{jk}&lt;/math&gt; to &lt;math&gt;{xy}+{xk}+{jy}&lt;/math&gt; in order to be able to factor it could be called &quot;completing the rectangle&quot; in analogy to the more familiar &quot;completing the square.&quot;<br /> <br /> == Applications ==<br /> This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are variables and &lt;math&gt;j,k&lt;/math&gt; are known constants. Also, it is typically necessary to add the &lt;math&gt;jk&lt;/math&gt; term to both sides to perform the factorization.<br /> <br /> == Problems ==<br /> ===Introductory===<br /> *Two different [[prime number]]s between &lt;math&gt;4&lt;/math&gt; and &lt;math&gt;18&lt;/math&gt; are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?<br /> <br /> &lt;math&gt; \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 } &lt;/math&gt;<br /> <br /> ([[2000 AMC 12/Problem 6|Source]])<br /> <br /> ===Intermediate===<br /> *&lt;math&gt;m, n&lt;/math&gt; are integers such that &lt;math&gt;m^2 + 3m^2n^2 = 30n^2 + 517&lt;/math&gt;. Find &lt;math&gt;3m^2n^2&lt;/math&gt;.<br /> <br /> ([[1987 AIME Problems/Problem 5|Source]])<br /> <br /> *The integer &lt;math&gt;N&lt;/math&gt; is positive. There are exactly &lt;math&gt;2005&lt;/math&gt; pairs &lt;math&gt;(x, y)&lt;/math&gt; of positive integers satisfying:<br /> <br /> &lt;cmath&gt;\frac 1x +\frac 1y = \frac 1N&lt;/cmath&gt;<br /> <br /> Prove that &lt;math&gt;N&lt;/math&gt; is a perfect square. (British Mathematical Olympiad Round 2, 2005)<br /> <br /> == See Also ==<br /> * [[Algebra]]<br /> * [[Factoring]]<br /> <br /> [[Category:Elementary algebra]]<br /> [[Category:Theorems]]</div> Ikitten https://artofproblemsolving.com/wiki/index.php?title=Simon%27s_Favorite_Factoring_Trick&diff=88271 Simon's Favorite Factoring Trick 2017-11-14T17:44:00Z <p>Ikitten: /* The General Statement */</p> <hr /> <div><br /> ==About==<br /> '''Dr. Simon's Favorite Factoring Trick''' (abbreviated '''SFFT''') is a special factorization first popularized by [[AoPS]] user [[user:ComplexZeta | Simon Rubinstein-Salzedo]].<br /> <br /> ==The General Statement==<br /> The general statement of SFFT is: &lt;math&gt;{xy}+{xk}+{jy}+{jk}=(x+j)(y+k)&lt;/math&gt;. Two special common cases are: &lt;math&gt;xy + x + y + 1 = (x+1)(y+1)&lt;/math&gt; and &lt;math&gt;xy - x - y +1 = (x-1)(y-1)&lt;/math&gt;.<br /> <br /> The act of adding &lt;math&gt;{jk}&lt;/math&gt; to &lt;math&gt;{xy}+{xk}+{yj}&lt;/math&gt; in order to be able to factor it could be called &quot;completing the rectangle&quot; in analogy to the more familiar &quot;completing the square.&quot;<br /> <br /> == Applications ==<br /> This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are variables and &lt;math&gt;j,k&lt;/math&gt; are known constants. Also, it is typically necessary to add the &lt;math&gt;jk&lt;/math&gt; term to both sides to perform the factorization.<br /> <br /> == Problems ==<br /> ===Introductory===<br /> *Two different [[prime number]]s between &lt;math&gt;4&lt;/math&gt; and &lt;math&gt;18&lt;/math&gt; are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?<br /> <br /> &lt;math&gt; \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 } &lt;/math&gt;<br /> <br /> ([[2000 AMC 12/Problem 6|Source]])<br /> <br /> ===Intermediate===<br /> *&lt;math&gt;m, n&lt;/math&gt; are integers such that &lt;math&gt;m^2 + 3m^2n^2 = 30n^2 + 517&lt;/math&gt;. Find &lt;math&gt;3m^2n^2&lt;/math&gt;.<br /> <br /> ([[1987 AIME Problems/Problem 5|Source]])<br /> <br /> *The integer &lt;math&gt;N&lt;/math&gt; is positive. There are exactly &lt;math&gt;2005&lt;/math&gt; pairs &lt;math&gt;(x, y)&lt;/math&gt; of positive integers satisfying:<br /> <br /> &lt;cmath&gt;\frac 1x +\frac 1y = \frac 1N&lt;/cmath&gt;<br /> <br /> Prove that &lt;math&gt;N&lt;/math&gt; is a perfect square. (British Mathematical Olympiad Round 2, 2005)<br /> <br /> == See Also ==<br /> * [[Algebra]]<br /> * [[Factoring]]<br /> <br /> [[Category:Elementary algebra]]<br /> [[Category:Theorems]]</div> Ikitten