Question #297636

Show that any prime number is either in form of 4k+1 or 4k+3, k is any positive integer


1
Expert's answer
2022-02-14T18:35:42-0500

By Euclid’s Division Lemma, each integer number nn can be represent as n=4k+r,n=4k+r, where r{0,1,2,3}.r\in\{0,1,2,3\}.

If r=0,r=0, then n=4k=2(2k),n=4k=2(2k), and hence nn is a composite number.

If r=2,r=2, then n=4k+2=2(2k+1),n=4k+2=2(2k+1), and hence nn is a composite number.

Therefore, any prime number is either in form of 4k+14k+1 or 4k+3,4k+3, where kk is any positive integer.


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