Question

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

Answer 1

Given: Let n be an arbitrary positive odd integer.

By Euclid’s division lemma divide n by 4: n = 4q + r with integer q and 0 ≤ r < 4, hence n = 4q or 4q + 1 or 4q + 2 or 4q + 3. 

Since n is odd, r cannot be 0 or 2, so r = 1 or 3; therefore n = 4q + 1 or n = 4q + 3.

Every positive odd integer is of the form 4q + 1 or 4q + 3 for some integer q.