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Prove-the-set-1-2-3-1989-can-be-expressed-as-the-disjoint-union-of-A-1-A-2-A-117-such-that-i-each-A-i-contains-the-same-number-of-elements-and-ii-the-sum-of-all-elements-of-each-A




Question Number 133419 by liberty last updated on 22/Feb/21
Prove the set {1,2,3,...,1989}  can be expressed as the disjoint  union of A_1 ,A_2 ,...,A_(117)  such that  (i) each A_i  contains the same number of elements ,and  (ii) the sum of all elements of each A_i  is  the same for i=1,2,3,...,m
$$\mathrm{Prove}\:\mathrm{the}\:\mathrm{set}\:\left\{\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{1989}\right\} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{expressed}\:\mathrm{as}\:\mathrm{the}\:\mathrm{disjoint} \\ $$$$\mathrm{union}\:\mathrm{of}\:\mathrm{A}_{\mathrm{1}} ,\mathrm{A}_{\mathrm{2}} ,…,\mathrm{A}_{\mathrm{117}} \:\mathrm{such}\:\mathrm{that} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{each}\:\mathrm{A}_{\mathrm{i}} \:\mathrm{contains}\:\mathrm{the}\:\mathrm{same}\:\mathrm{number}\:\mathrm{of}\:\mathrm{elements}\:,\mathrm{and} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{elements}\:\mathrm{of}\:\mathrm{each}\:\mathrm{A}_{\mathrm{i}} \:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{for}\:\mathrm{i}=\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{m} \\ $$

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