Show that every positive odd integer is of the form (4q + 1) or (4q + 3), where q is some integer.

Let a be any positive odd integer and b = 4. So by applying division lemma to a, b
a = 4q + r where 0 ≤ r < 4
When r = 0, a = 4q + 0 = 4q = 2(2q)
When r = 1, a = 4q + 1
When r = 2, a = 4q + 2 = 2(2q + 1)
When r = 3, a = 4q + 3
Since 4q, 4q + 2 are multiples of 2, so they are even numbers.
So every positive odd integer is of the form (4q + 1) or (4q + 3) where q is some integer