Brain Teaser - Working with numbers that don't divide into one another?

1 Answer

Answer :

1) Write all of those numbers in the form: 2^k * m where m is an odd integer and k is a non-negative integer. Obviously m < 50. Since we have 25 odd numbers less than 50 if we pick 26 numbers between 1 and 50 two of these m’s are the same. Therefore two of these numbers are of the form x = 2^k * m and y = 2^l * m. Now if k < l then x divides y. If k > l then y divides x. 2) If we have a selection of 25 numbers that don’t divide one another, then by the above argument m’s should be all odd numbers less than 50. Now, for any odd number m less than 50 pick k to be the largest number that satisfies 2^k * m < = 50. Obviously if I show this set satisfies the condition, this is the largest possible set. The set satisfies the condition, because if 2^k * m divides 2^l * n then n is a multiple of m, and hence n is at least 3 m. Hence k is larger than l. For (3) I should think a bit more.

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