Question

Using Euclids division lemma show that any positive odd integer in the form of 6q+1,6q+3,6q+5.where q is some integer

Answer 1

Let be a given integer.
On dividing a by 6, we get q as the quotient and r as remainder such that
a=6q +r, r=0,1,2,3,4,5
when r =0
a =6q, even no
when r =1
a =6q +1, odd no
when r =2
a=6q +2, even no
when r =3
a=6q+3, odd no
when r =4
a =6q+4, even no
when r =5,
a=6q+5, odd no
$\therefore$ Any positive odd integer is of the form 6q+1.6q+3 or 6q +5