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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
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)
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