Show that every positive odd integer is of the form (4q + 1) or (4q + 3) for some integer q.
Let a be any odd positive integer and b = 4.
By division lemma there exist integer q and r such that
a = 4 q + r, where 0 ≤ r ≤ 4
so a = 4q or, a = 4q + 1 or, a = 4q + 2 or, a = 4q + 3
4q + 1 4 is divisible by 2 but 1 is not divisible by 2,so it is an odd number
4q + 2 4 is divisible by 2 and 2 is also divisible by 2,so it is an even number
4q + 3 4 is divisible by 2 but 3 is not divisible by 2,so it is an odd number
4q + 4 4 is divisible by 2 and 4 is also divisible by 2,so it is an even number
∴, any odd integer is of the form 4q + 1 or, 4q + 3.