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May 24th, 2011, 05:26 PM  #1 
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Least integer having all digits 4 divisible by 169
(Olimpíada de Mayo, 2010)  Find the smallest integer of the form 4444.. (all digits 4) which is divisible by 169. My progress: If k is the number of digits in the number, then obviously , so it can be found that k must be divisible by 6. Using any computational tools, it is readily shown that k = 78, but since this is an olympiad question, it should be done by hand. Should I list all the possibilities for mod 169, for k = 6, 12, ... 78? There should be a simpler way. Additional question: Is it possible to define the length of the period of for any p without listing the remainders? 
May 24th, 2011, 09:47 PM  #2 
Newbie Joined: May 2011 Posts: 19 Thanks: 0  Re: Least integer having all digits 4 divisible by 169 and divides it only if divides and divides that only if divides , so . Euler's totient so is the smallest divisor such that . There are possibilities to check in order of size: * * * * * * we stop here because we found the answer. 
May 25th, 2011, 09:30 AM  #3 
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Re: Least integer having all digits 4 divisible by 169
Thanks! so I guess that this is true: 
May 25th, 2011, 10:11 AM  #4 
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Re: Least integer having all digits 4 divisible by 169
Sorry, my last post's statement is obviously wrong. But why must k be a divisor of phi(13^2) so that 10^k is congruent to 1 modulo 13²?

May 25th, 2011, 10:42 AM  #5 
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  Re: Least integer having all digits 4 divisible by 169
The period repeating decimals of 1/169 is 78. I think that too indicates that the amount of 4's is 78. http://en.wikipedia.org/wiki/Repeating_ ... rime_to_10 
May 25th, 2011, 06:38 PM  #6  
Newbie Joined: May 2011 Posts: 19 Thanks: 0  Re: Least integer having all digits 4 divisible by 169 Quote:
 

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