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2004 AMC 12A Problems/Problem 25

Revision as of 10:27, 4 December 2007 by 1=2 (talk | contribs) (LaTeX issue fixed)

Problem

For each integer $n\geq 4$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. The product $a_4a_5...a_{99}$ can be expressed as $\frac {m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$?

$\text {(A)} 98 \qquad \text {(B)} 101 \qquad \text {(C)} 132\qquad \text {(D)} 798\qquad \text {(E)}962$

Solution

\[a_x = \frac{1}{x}+\frac{3}{x^2}+\frac{3}{x^3}+\frac{1}{x^4}+\frac{3}{x^5}+\frac{3}{x^6}+\cdots\]

\[a_x*x^3=x^2+3x+3+a_x\]

\[a_x(x^3-1)=x^2+3x+3\]

\[a_x=\frac{x^2+3x+3}{x^3-1}=\frac{(x+1)^3-1}{x(x^3-1)}\]


$a_4a_5...a_{99}=\frac{(5^3-1)(6^3-1)\cdots (100^3-1)}{4*5*6*\cdots*99*(4^3-1)(5^3-1)\cdots(99^3-1)}$

$a_4a_5...a_{99}=\frac{999999}{4*5*6*\cdots*99*63}=\frac{13*37*33*6}{99!}$

Since $13*37*33*6$ isn't one of the answer choices, we need to get rid of some stuff:

$99=33*3$

$\frac{13*37*33*6}{99!}=\frac{13*37*2}{98!}$

Since only the two goes into 98, n is at it's minimum.

$13*37*2=962 \Rightarrow \text {(E)}$

See Also

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