Pick up Ten
What is the maximum number of positive integers we could pick, such that
1) the absolute difference of any two of them is at most 9, and
2) any two of them are coprime.
For example, the numbers shows that the maximum is at least 4.
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1 solution
Discussions for this problem are now closed
The absolute difference is at most 9, means that we are looking at sets of 10 consecutive positive integers.
Out of 10 consecutive positive integers, 5 of them will be even. We can pick at most 1 out of these 5 even numbers, and thus at most 6 out of these 10 consecutive numbers.
Consider the sequence 1 1 , 1 2 , 1 3 , 1 4 , … , 1 9 . We can pick 1 1 , 1 3 , 1 5 , 1 6 , 1 7 , 1 9 in which any 2 of them are coprime. Hence, the answer is exactly 6.