Question

Show that positive integer of the form 3q or 3q +1 or 3q+2 for some integer q

Answer 1

any positive integer n can be written as 3q , 3q+1 , 3q+2
if n=3q
n square=9 square =3 (3q)=3m where m=3q
if n= 3q+1
n square = (3q+1)Square =9 square +6q+2=3 (3q square +2q)+1=3m+1 where m=3q square + 2q
same with 3 Rd one
and at last n=3q or 3q+1 but not for 3q+2q



may it help you☺

Answer 2

It is very easy
Let n be an arbitrary positive intiger
Then dividing n by 3 let q be the quotiont and r be the remainder
Then, by euclids lemma,
n=3q+r. Where(0<=r<3)
n= 3q, n=3q+1, or n=3q+2.......... For some integer q
Thus any positive integer in the form 3q, 3q+1,or3q+2 for some integer q,.