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In every (n + 1) - - elementic subset of the set (1, 2, 3, .......2n) which of the following is correct:
- a)exist at least three natural number which are prime to each other
- b)There exist no consecutive natural number
- c)There exist at least two natural numbers which are prime to each other
- d)There exist more than two natural numbers which are prime to each other
- e)None of the above
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
In every (n + 1) - - elementic subset of the set (1, 2, 3, .......2n) ...
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.
Community Answer
In every (n + 1) - - elementic subset of the set (1, 2, 3, .......2n) ...
Explanation:
Given set: (1, 2, 3, .......2n)
a) Existence of at least three natural numbers which are prime to each other:
- In a set of (n-1) elements, it is not guaranteed that there will be at least three numbers that are prime to each other. This cannot be generalized for all subsets.
b) Absence of consecutive natural numbers:
- While it is true that consecutive natural numbers do not exist in a (n-1) element subset, this does not guarantee the existence of prime numbers.
c) Existence of at least two natural numbers which are prime to each other:
- By the Pigeonhole Principle, in any subset of (n-1) elements taken from a set of 2n elements, there will be at least one pair of numbers that are relatively prime to each other.
d) Existence of more than two natural numbers which are prime to each other:
- The question does not specify that there will be more than two numbers that are prime to each other in any given subset. Therefore, this statement cannot be concluded.
e) None of the above:
- The correct answer is option 'C' because the existence of at least two natural numbers that are prime to each other can be guaranteed in any (n-1) element subset of the given set.
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In every (n + 1) - - elementic subset of the set (1, 2, 3, .......2n) which of the following is correct:a)exist at least three natural number which are prime to each otherb)There exist no consecutive natural numberc)There exist at least two natural numbers which are prime to each otherd)There exist more than two natural numbers which are prime to each othere)None of the aboveCorrect answer is option 'C'. Can you explain this answer?
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