Problem 7. 40 points) In this problem, you will prove that division with remainder is well defined. That is, you will prove that for any two integers a 0 and b, there exists a unique remainder after the division of b and 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 , but don't assume anything about integer division. In particular, you can use the fact that there are no integer multiplies of a that are greater than 0 and less than a (b) (20 points) Let S = { integer s > 0 : integer q such that b= aq + s). You can use without proof the following fact: every nonempty subset of nonnegative integers contains an element that is smaller than all other values in the subset. Prove that S contains a remainder after the division of b by a (that is, there is at least one remainder)