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.
(In reply to re(2): The solution is in the problem.==>not so
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