Show that any positive odd integer is of the form (4m+1) or (4m+3), where m is some integer.

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This is the basic and conceptual question from real numbers in which we have been asked to prove that any positive odd integer is of the form (4m+1) or (4m+3), where m is some integer.

## LetÂ sÂ be any positive integer.

On dividingÂ sÂ byÂ 4,Â letÂ mÂ be the quotient andÂ rÂ be the remainder

By Euclid’s division lemma,

s=4m+r,Â whereÂ 0â‰¤r<4

So we haveÂ ,s=4mÂ orÂ s=4m+1Â orÂ s=4m+2Â orÂ s=4m+3.

hereÂ 4m,4m+2Â are multiples ofÂ 2,Â which revert to even values ofÂ s.

again ,Â s=4m+1Â orÂ s=4m+3Â are odd values ofÂ s.

Thus, any positive odd integer is of the formÂ (4m+1)Â orÂ (4m+3)Â whereÂ sÂ is any odd integer.