Let A = {2, 3, . . . , 50}, that is, A is the set of positive integers greater than 1 and less than 51. Determine the smallest number x such that every subset of A having x elements contains at least two integers that have a common divisor greater than 1, and justify your answer.
Homework Answers
For having common divisor integers should not be co-prime.
Prime numbers in set A are = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Count of prime numbers = 15
For determining smallest number x such that every subset of A having x elements contains at least two integers that have a common divisor greater than 1, we need to take worst case scenario.
If subset of A has all prime number the above condition won't be fulfil.
So we need to have just one more number than count of prime numbers in the subset.
Thus x = 16
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1) positive, greater than 0
OR negative, less than 0
2) positive, greater than 0 OR negative, less
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3) positive, greater than 0 OR negative, less
than 0
4) positive, greater than 0 OR negative, less
than 0
5) smaller, larger, smaller or larger depending on temp.
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