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26 out of 50 (Posted on 2016-11-25) Difficulty: 3 of 5
Prove the following statement:

In any set of 26 integers chosen from the set of (1,2,3, ...50) there must be at least a pair of numbers such that one of them divides the other.

Generalize.

No Solution Yet Submitted by Ady TZIDON    
Rating: 4.0000 (1 votes)

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Hints/Tips re(3): The solution is in the problem.==>not so | Comment 4 of 10 |
(In reply to re(2): The solution is in the problem.==>not so by broll)

It is not gilding the lily.

 Both of us agree that 26 to 50 fits the description of a largest (25 members) set where no pair of numbers has a common divisor.
Clearly, this not the only set: e.g. 
replacing some of the semiprime numbers (=p*q) - with some restrictions - produces qualifying sets.
Example: Example:replace 34 by  17, 42 by 21, replace 26 or 39 (but not both!) by 13  etc.
It applies not only to semiprime numbers,  pq^2 might be replaced by pq or by q^2 in some cases (50  by 25,  44 by 22    etc)...
If you want, you may follow from here to provide a solid proof - there not too many cases to consider.



Edited on November 27, 2016, 9:19 pm
  Posted by Ady TZIDON on 2016-11-26 02:20:07

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