question_answer

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

Answer:

    Let a be an odd integer and b = 4. By division algorithm, there exists integers q and r such that , where   On putting r = 0, 1, 2 and 3 we get a   = 4q, 4q + 1, 4q + 2 and 4q + 3 But a is an odd integer Hence a = 4q + 1 or a = 4q + 3. Hence the result.