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

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Question Number 133419 by liberty last updated on 22/Feb/21

$$\mathrm{Prove}\mathrm{the}\mathrm{set}\{1,2,3,\dots ,1989\}$$$$\mathrm{can}\mathrm{be}\mathrm{expressed}\mathrm{as}\mathrm{the}\mathrm{disjoint}$$$$\mathrm{union}\mathrm{of}{\mathrm{A}}_1,{\mathrm{A}}_2,\dots ,{\mathrm{A}}_{117}\mathrm{such}\mathrm{that}$$$$\left(\mathrm{i}\right)\mathrm{each}{\mathrm{A}}_{\mathrm{i}}\mathrm{contains}\mathrm{the}\mathrm{same}\mathrm{number}\mathrm{of}\mathrm{elements},\mathrm{and}$$$$\left(\mathrm{ii}\right)\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{all}\mathrm{elements}\mathrm{of}\mathrm{each}{\mathrm{A}}_{\mathrm{i}}\mathrm{is}$$$$\mathrm{the}\mathrm{same}\mathrm{for}\mathrm{i}=1,2,3,\dots ,\mathrm{m}$$

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