https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Robotik&feedformat=atomAoPS Wiki - User contributions [en]2024-03-19T13:48:10ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2000_AMC_12_Problems/Problem_6&diff=1280762000 AMC 12 Problems/Problem 62020-07-11T18:11:18Z<p>Robotik: /* Problem */</p>
<hr />
<div>{{duplicate|[[2000 AMC 12 Problems|2000 AMC 12 #6]] and [[2000 AMC 10 Problems|2000 AMC 10 #11]]}}<br />
<br />
==Solution 1==<br />
<br />
Any two prime numbers between 4 and 18 have an odd product and an even sum. Any odd number minus an even number is an odd number, so we can eliminate B, D, and A. Since the highest two prime numbers we can pick are 13 and 17, the highest number we can make is <math>(13)(17)-(13+17) = 221 - 30 = 191</math>. Thus, we can eliminate E. So, the answer must be <math>\boxed{\textbf{(C) }119}</math>.<br />
<br />
==Solution 2==<br />
<br />
Let the two primes be <math>p</math> and <math>q</math>. We wish to obtain the value of <math>pq-(p+q)</math>, or <math>pq-p-q</math>. Using [[Simon's Favorite Factoring Trick]], we can rewrite this expression as <math>(1-p)(1-q) -1</math> or <math>(p-1)(q-1) -1</math>. Noticing that <math>(13-1)(11-1) - 1 = 120-1 = 119</math>, we see that the answer is <math>\boxed{\textbf{(C) }119}</math>.<br />
<br />
==Solution 3==<br />
The answer must be in the form <math>pq - p - q</math> = <math>(p - 1)(q - 1) - 1</math>. Since <math>p - 1</math> and <math>q - 1</math> are both even, <math>(p - 1)(q - 1) - 1</math> is <math>3 \pmod 4</math>, and the only answer that is <math>3 \pmod 4</math> is <math>\boxed{\textbf{(C) }119}</math>.<br />
<br />
==Videos:==<br />
https://www.youtube.com/watch?v=ddE5GO1RNLw&t=1s<br />
<br />
== See also ==<br />
{{AMC10 box|year=2000|num-b=10|num-a=12}}<br />
{{AMC12 box|year=2000|num-b=5|num-a=7}}<br />
[[Category:Introductory Algebra Problems]]<br />
{{MAA Notice}}</div>Robotikhttps://artofproblemsolving.com/wiki/index.php?title=Simon%27s_Favorite_Factoring_Trick&diff=128075Simon's Favorite Factoring Trick2020-07-11T18:10:06Z<p>Robotik: /* Fun Practice Problems */</p>
<hr />
<div><br />
==The General Statement==<br />
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>.<br />
<br />
The act of adding <math>{jk}</math> to <math>{xy}+{xk}+{yj}</math> in order to be able to factor it could be called "completing the rectangle" in analogy to the more familiar "completing the square."<br />
<br />
== Applications ==<br />
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.<br />
<br />
== Fun Practice Problems ==<br />
===Introductory===<br />
*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?<br />
<br />
<math> \mathrm{(A) \ 22 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 } </math><br />
<br />
([[2000 AMC 12/Problem 6|Source]])<br />
<br />
===Intermediate===<br />
*<math>m, n</math> are integers such that <math>m^2 + 3m^2n^2 = 30n^2 + 517</math>. Find <math>3m^2n^2</math>.<br />
<br />
([[1987 AIME Problems/Problem 5|Source]])<br />
<br />
===Olympiad===<br />
<br />
*The integer <math>N</math> is positive. There are exactly 2005 ordered pairs <math>(x, y)</math> of positive integers satisfying:<br />
<br />
<cmath>\frac 1x +\frac 1y = \frac 1N</cmath><br />
<br />
Prove that <math>N</math> is a perfect square. (British Mathematical Olympiad Round 3, 2005)<br />
<br />
== See More==<br />
* [[Algebra]]<br />
* [[Factoring]]<br />
<br />
[[Category:Elementary algebra]]<br />
[[Category:Theorems]]</div>Robotik