Show the every positive odd integer is of the form ( 6m+1) or ( 6m+3) or ( 6m+5) for some intger m.

Explanation

Let n be a given positive odd interger.
On dividings n by 6, let m be the quotient and r be the remainder.then, by Euclid's division lemma, we have
n = 6m + r, where ` 0 le r lt 6`
` Rightarrow n= 6m +r, r "where" r=0,1,2,3,4,5`
`Rightarrow n=6m or ( 6m +1 ) or ( 6m+2) or ( 6m+3) or ( 6m+4) or ( 6m +2)`
But , n=6 m, ( 6m +2) ( 6m+4) given ever values of n.
thus, when n is odd, it is of the form ( 6m+1) or ( 6m+3) or ( 6m+5) for some integer m.