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Let-A-be-a-set-of-16-positive-integers-with-the-property-that-the-product-of-any-two-distinct-numbers-of-A-will-not-exceed-1994-Show-that-there-are-two-numbers-a-and-b-in-A-which-are-not-relatively-p




Question Number 21962 by Tinkutara last updated on 07/Oct/17
Let A be a set of 16 positive integers  with the property that the product of  any two distinct numbers of A will  not exceed 1994. Show that there are  two numbers a and b in A which are  not relatively prime.
$$\mathrm{Let}\:{A}\:\mathrm{be}\:\mathrm{a}\:\mathrm{set}\:\mathrm{of}\:\mathrm{16}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{property}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of} \\ $$$$\mathrm{any}\:\mathrm{two}\:\mathrm{distinct}\:\mathrm{numbers}\:\mathrm{of}\:{A}\:\mathrm{will} \\ $$$$\mathrm{not}\:\mathrm{exceed}\:\mathrm{1994}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are} \\ $$$$\mathrm{two}\:\mathrm{numbers}\:{a}\:\mathrm{and}\:{b}\:\mathrm{in}\:{A}\:\mathrm{which}\:\mathrm{are} \\ $$$$\mathrm{not}\:\mathrm{relatively}\:\mathrm{prime}. \\ $$
Commented by Rasheed.Sindhi last updated on 08/Oct/17
Possible set of maximum length  (i)whose elements are pairwise   coprime and  (ii)whose product of any two    elements does not exceed 1994  is of length 14 and is as follows:  {2,3,5,7,11,13,17,19,23,29,31,37,41,43}
$$\mathrm{Possible}\:\mathrm{set}\:\mathrm{of}\:\mathrm{maximum}\:\mathrm{length} \\ $$$$\left(\mathrm{i}\right)\mathrm{whose}\:\mathrm{elements}\:\mathrm{are}\:\mathrm{pairwise}\: \\ $$$$\mathrm{coprime}\:\mathrm{and} \\ $$$$\left(\mathrm{ii}\right)\mathrm{whose}\:\mathrm{product}\:\mathrm{of}\:\mathrm{any}\:\mathrm{two}\: \\ $$$$\:\mathrm{elements}\:\mathrm{does}\:\mathrm{not}\:\mathrm{exceed}\:\mathrm{1994} \\ $$$$\mathrm{is}\:\mathrm{of}\:\mathrm{length}\:\mathrm{14}\:\mathrm{and}\:\mathrm{is}\:\mathrm{as}\:\mathrm{follows}: \\ $$$$\left\{\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{7},\mathrm{11},\mathrm{13},\mathrm{17},\mathrm{19},\mathrm{23},\mathrm{29},\mathrm{31},\mathrm{37},\mathrm{41},\mathrm{43}\right\} \\ $$
Commented by Tinkutara last updated on 08/Oct/17
Thank you very much Sir! Now solved.
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}!\:\mathrm{Now}\:\mathrm{solved}. \\ $$

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