My Math Forum Cyclical Permutation Problem

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 August 31st, 2010, 11:29 AM #1 Newbie   Joined: May 2007 Posts: 18 Thanks: 0 Cyclical Permutation Problem Hey guys! Can one of you help me with the following problem? Prove that if a number of n digits, expressed in the scale of r, is divisible by any factor of r^[n-1], that divisibility is not altered by a cyclical permutation of the digits of the original number. Thank!
August 31st, 2010, 01:40 PM   #2
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 408

Re: Cyclical Permutation Problem

Hello, morning_mood!

Please check the wording of the problem.
As stated, the claim is not true.

Quote:
 $\text{Prove that if a number of }n\text{ digits, expressed in the scale of }r$ $\text{is divisible by any factor of }r^{n-1},\,\text{ that divisibility is not altered}$ $\text{by a cyclical permutation of the digits of the original number.}$
$\text{Let }n \,=\,3 \;\;\;\cdots\;\;\text{We have a 3-digit number, }N.$

$\text{Let }r\,=\,10\;\;\cdots\;\;N\text{ is expressed in base 10.}$

$\text{And }N\text{ is divisible by any factor of }10^2 \:=\:2^2\,\cdot\,5^2\;\;\cdots\;\text{ say, 5.}$

$\text{So, }N\text{ could be }175\text{, which is divisible by 5.}$

$\text{It is claimed that }751\text{ and }517\text{ are divisible by 5.}$
[color=beige]. . . . . . . . . . . . . . [/color][color=red]? ? ?[/color]

 August 31st, 2010, 04:31 PM #3 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Cyclical Permutation Problem I'm guessing it should be r^n - 1, not r^(n - 1).
 August 31st, 2010, 07:49 PM #4 Math Team   Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408 Re: Cyclical Permutation Problem The problem is similar to a phenomenon in base-ten. If a number $N$ is divisible by 9, then any cyclic permutation [color=beige]. . [/color] of the digits of $N$ is also divisible by 9.
 September 1st, 2010, 04:29 AM #5 Newbie   Joined: May 2007 Posts: 18 Thanks: 0 Re: Cyclical Permutation Problem You guys are right, it should have been r^1 - 1. I got confused about how write powers here and reported the problem incorrectly. Thanks for the help!

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