Question

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

Answer 1

Let a be the given positive odd integer.
On dividing a by 4,let q be the quotient and r the remainder.
Therefore,by Euclid's algorithm we have
a = 4q + r 0 ≤ r < 4
⇒ a = 4q + r r​ = 0,1,2,3
⇒ a = 4q, a = 4q + 1, a = 4q + 2, a = 4q + 3
But, 4q and 4q + 2 = 2 (2q + 1) = even
Thus, when a is odd, it is of the form (4q + 1) or (4q + 3) for some integer q.