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Primes vs ones (Posted on 2014-03-27) Difficulty: 3 of 5
Prove - or find a counterexample:

All primes p greater than 5 divide a number
whose decimal representation consists of (p-1) ones.

No Solution Yet Submitted by Ady TZIDON    
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Another proof Comment 2 of 2 |
List the first p rep-units beginning with the smallest rep-unit >= p.  

Divide them all by p.  The p remainders will be in the list 0,1,2, . . . , (p-1).  

If any remainder = 0, we're finished.  

Otherwise, we have p remainders chosen from the remaining (p-1) possibilities so some remainders will be duplicates.  

Choose two rep-units with the same remainder.  Call them x and y.  

Then (x-y) is divisible by p since it has a remainder of 0 after division by p.  But (x-y) = (power of ten) times (rep-unit).  Given that p>5 it can't factor (power of ten) so it must factor (rep-unit).  

Edited on March 28, 2014, 4:03 pm
  Posted by xdog on 2014-03-28 16:01:57

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