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 Not So Neighborly (Posted on 2004-03-03)
Prove that at least one integer in any set of ten consecutive integers is relatively prime to the others in the set.

 Submitted by Aaron Rating: 4.0000 (4 votes) Solution: (Hide) There are 5 odd numbers in the set. At most 2 are multiples of 3, at most 1 is a multiple of 5 and at most 1 is a multiple of 7. So, there is at least one odd number, k, that is not divisible by 3, 5 or 7. Now if k has a common factor with another member in the set, then that factor must divide their difference, which is at most 9. However, the common factor cannot be divisible by 2, 3, 5 or 7, so it must be 1.

 Subject Author Date This meeting of the google fanclub is called to order. :-) Penny 2004-03-03 13:04:36 Extension in the other direction Charlie 2004-03-03 09:50:20 Another proof--and extension. Charlie 2004-03-03 09:07:57 re(2): ELEMENTARY, charlie Ady TZIDON 2004-03-03 08:45:12 re: ELEMENTARY, DEAR WATSON Charlie 2004-03-03 08:39:55 solution Charlie 2004-03-03 08:37:43 ELEMENTARY, DEAR WATSON Ady TZIDON 2004-03-03 08:32:02

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