Question

show the positive odd into her is of the form 6q+1 or 6q+3 or 6q+5 where q is some intiger

Answer 1

let p be a given positive odd integer.
on dividing p by 6, let q be the quotient and r be the remainder
then, by Euclid division lemma , we have
p= 6q+ r, where 0 < r < 6
p= 6q+r , where r = 0,1,2,3,4,5
p= 6q or (6q+1 )or (6q +2) or (6q +3) or (6q+4 )or (6q+5)
but, p= 6q or (6q+2 )or (6q +4) gives even value of p
thus , when p is odd , it is of the form (6q+1 )or (6q+3) or (6q+ 5) for some integer q