Show that any positive odd integer of the form 6q+ 1 or 6Q + 3 or 6Q + 5 were q in some integer

Let a be any positive odd integer and b = 6.

We can apply Euclid’s division algorithm on a and b = 6.

⇒a=6q+r

We know that value of b = 6.

⇒0≤r<6

∴ All possible values of a are:

a = 6q
a = 6q+1
a = 6q+2
a = 6q+3
a = 6q+4
a = 6q+5

Here 6q, 6q+2 and 6q+4 are divisible by 2 since 2 is factor of them.

⇒ They are not positive odd integers.

∴ 6q+1, 6q+3 and 6q+5 are positive odd integers.

∴ Any positive odd integer is of the form (6q+1) or (6q+3) or (6q+5).