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Question Number 44059 by paro123 last updated on 20/Sep/18

$$\mathrm{Prove}\mathrm{that}\mathrm{any}\mathrm{integer}\mathrm{can}\mathrm{be}\mathrm{expressed}$$$$\mathrm{as}\mathrm{in}\mathrm{the}\mathrm{form}\mathrm{of}4\mathrm{k}\mathrm{or}4\mathrm{k}\underset{-}{+}1\mathrm{or}4\mathrm{k}\underset{-}{+}2.$$

Answered by kunal1234523 last updated on 21/Sep/18

$$anyintegerpcanbeexpressedintheform$$$$p=4q\pm r$$$$here,r=0,1,2,3$$$$case1$$$$r=0$$$$p=4q\pm 0=4q$$$$similarlyfor$$$$r=1p=4q\pm 1$$$$r=2p=4q\pm 2$$$$forr=3$$$$p=4q\pm 3$$$$p=4q\pm 4\mp 1$$$$p=4(q\pm 1)\mp 1$$$$p=4m\mp 1=4m\pm 1,m=q\pm 1$$$$thisissameforr=1$$$$sop=4q,4q\pm 1,4q\pm 2$$

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