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Not So Neighborly (Posted on 2004-03-03) Difficulty: 4 of 5
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.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
This meeting of the google fanclub is called to order. :-)Penny2004-03-03 13:04:36
Extension in the other directionCharlie2004-03-03 09:50:20
SolutionAnother proof--and extension.Charlie2004-03-03 09:07:57
re(2): ELEMENTARY, charlieAdy TZIDON2004-03-03 08:45:12
re: ELEMENTARY, DEAR WATSONCharlie2004-03-03 08:39:55
SolutionsolutionCharlie2004-03-03 08:37:43
SolutionELEMENTARY, DEAR WATSONAdy TZIDON2004-03-03 08:32:02
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