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Question Number 21784 by Tinkutara last updated on 03/Oct/17

Call a positive integer n good if there are n integers, positive or negative, and not necessarily distinct, such that their sum and product are both equal to n (e.g. 8 is good since 8=4∙2∙1∙1∙1∙1(−1)(−1)=4+2+1+1+1 +1+(−1)+(−1)). Show that integers of the form 4k + 1 (k ≥ 0) and 4l (l ≥ 2) are good.

$$\mathrm{Call}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer}\:{n}\:\boldsymbol{\mathrm{good}}\:\mathrm{if}\:\mathrm{there} \\ $$$$\mathrm{are}\:{n}\:\mathrm{integers},\:\mathrm{positive}\:\mathrm{or}\:\mathrm{negative},\:\mathrm{and} \\ $$$$\mathrm{not}\:\mathrm{necessarily}\:\mathrm{distinct},\:\mathrm{such}\:\mathrm{that}\:\mathrm{their} \\ $$$$\mathrm{sum}\:\mathrm{and}\:\mathrm{product}\:\mathrm{are}\:\mathrm{both}\:\mathrm{equal}\:\mathrm{to}\:{n} \\ $$$$\left(\mathrm{e}.\mathrm{g}.\:\mathrm{8}\:\mathrm{is}\:\boldsymbol{\mathrm{good}}\:\mathrm{since}\right. \\ $$$$\mathrm{8}=\mathrm{4}\centerdot\mathrm{2}\centerdot\mathrm{1}\centerdot\mathrm{1}\centerdot\mathrm{1}\centerdot\mathrm{1}\left(−\mathrm{1}\right)\left(−\mathrm{1}\right)=\mathrm{4}+\mathrm{2}+\mathrm{1}+\mathrm{1}+\mathrm{1} \\ $$$$\left.+\mathrm{1}+\left(−\mathrm{1}\right)+\left(−\mathrm{1}\right)\right). \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{integers}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form}\:\mathrm{4}{k}\:+\:\mathrm{1} \\ $$$$\left({k}\:\geqslant\:\mathrm{0}\right)\:\mathrm{and}\:\mathrm{4}{l}\:\left({l}\:\geqslant\:\mathrm{2}\right)\:\mathrm{are}\:\boldsymbol{\mathrm{good}}. \\ $$

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