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May 24th, 2011, 05:26 PM   #1
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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?
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May 24th, 2011, 09:47 PM   #2
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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.
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May 25th, 2011, 09:30 AM   #3
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Re: Least integer having all digits 4 divisible by 169

Thanks! so I guess that this is true:

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May 25th, 2011, 10:11 AM   #4
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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?
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May 25th, 2011, 10:42 AM   #5
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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
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May 25th, 2011, 06:38 PM   #6
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Re: Least integer having all digits 4 divisible by 169

Quote:
Originally Posted by proglote
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?
It's a consequence of group theory (Lagrange's theorem). The order of an element is the smallest natural number such that , every element has order dividing .
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