In this problem, you will prove that division with remainder is welldefined. That is, you will prove that for any two integers a > 0 and b, there exists a unique remainder after the division of b by a. We will do the proof in two parts: first by proving that there is at most one remainder, and then by proving that there exists at least one remainder. The conclusion will be that there exists a unique remainder. Assume a and b are both integers and a > 0. Define a remainder after the division of b by a to be a value r such that r ≥ 0, r < a, and there exists an integer q for which b = aq + r