Difference between revisions of "2004 AMC 12A Problems/Problem 25"
(New page: ==Problem== For each integer <math>n\geq 4</math>, let <math>a_n</math> denote the base-<math>n</math> number <math>0.\overline{133}_n</math>. The product <math>a_4a_5...a_{99}</math> can ...) |
(LaTeX issue fixed) |
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| − | <math>a_4a_5...a_{99}=\frac{(5^3-1)(6^3-1)\cdots (100^3-1)}{4*5*6*\cdots*99*(4^3-1)(5^3-1)\cdots(99^3-1)</math> | + | <math>a_4a_5...a_{99}=\frac{(5^3-1)(6^3-1)\cdots (100^3-1)}{4*5*6*\cdots*99*(4^3-1)(5^3-1)\cdots(99^3-1)}</math> |
<math>a_4a_5...a_{99}=\frac{999999}{4*5*6*\cdots*99*63}=\frac{13*37*33*6}{99!}</math> | <math>a_4a_5...a_{99}=\frac{999999}{4*5*6*\cdots*99*63}=\frac{13*37*33*6}{99!}</math> | ||
Revision as of 11:27, 4 December 2007
Problem
For each integer 
, let 
denote the base-
number 
. The product 
can be expressed as 
, where 
and are positive integers and is as small as possible. What is the value of ?
Solution
![\[a_x = \frac{1}{x}+\frac{3}{x^2}+\frac{3}{x^3}+\frac{1}{x^4}+\frac{3}{x^5}+\frac{3}{x^6}+\cdots\]](http://latex.artofproblemsolving.com/c/e/0/ce0d20f34e7ee1ab8c317aef53d1ae200e46a21d.png)
![\[a_x*x^3=x^2+3x+3+a_x\]](http://latex.artofproblemsolving.com/0/1/0/0101bd5ec2a8e93b0e3ceab413d400150b3da406.png)
![\[a_x(x^3-1)=x^2+3x+3\]](http://latex.artofproblemsolving.com/e/b/7/eb7f27fa0513208f0b475e567f045b350b210238.png)
![\[a_x=\frac{x^2+3x+3}{x^3-1}=\frac{(x+1)^3-1}{x(x^3-1)}\]](http://latex.artofproblemsolving.com/4/9/2/492ec8e41e19d177dd528d91e141a24de34d3aa3.png)
Since 
isn't one of the answer choices, we need to get rid of some stuff:
Since only the two goes into 98, n is at it's minimum.
