Question

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

Answer 1

Let a be any odd positive integer. we have to prove that a is of the form 4q + 1 or 4q + 3, where q is some integer.
Since a is an integer, consider b = 4 as another integer.
Applying Euclid's division lemma, we get:
a = 4q + r for some integer q ≥ 0 and r = 0, 1, 2 and 3, since 0 ≤ r < 4.
Therefore, a = 4q or 4q + 1 or 4q + 2 or 4q + 3.
However, since a is odd, it cannot take the values 4q or 4q + 2 (since all these are divisible by 2).
Hence, any odd integer can be expressed in the form 4q + 1 or 4q + 3, where q is some integer.