Question

In every (n + 1) - - elementic subset of the set (1, 2, 3, .......2n) which of the following is correct:
1. There exist at least two natural numbers which are prime to each other
2. exist at least three natural umber which are prime to each other
3. There exist no consecutive natural number
4. There exist more than two natural numbers which are prime to each other

Answer 1

Correct Answer - Option 1 : There exist at least two natural numbers which are prime to each other

Concept:

The Pigeonhole Principle: Let there be n boxes and (n + 1) objects. Then, under any assignment of objects to the boxes, there will always be a box with more than one object in it. This can be reworded as, if m pigeons occupy n pigeonholes, where m > n, then there is at least one pigeonhole with two or more pigeons in it. 

Calculation:        

We divide the set into n classes {1, 2}, {3, 4},......{2n - 1, 2n}.

By the pigeonhole principle, given n +1 elements at least two of them will be in the same case {2k - 1, 2k} (1 ≤ k ≤ n). But 2k - 1 and 2k are relatively prime because their difference is 1.