**15**

The division theorem

The remainder of this book illustrates how the methods of proof and the language of sets and functions introduced so far are applied in arithmetic, the theory of numbers. We begin in this chapter by discussing the elementary ideas of division and remainders which we all met at an early stage of our schooling. The formal treatment of this generalizes the discussion of odd and even integers in Section 11.3.

The main result of this chapter is the following.

**Theorem 15.1.1 (The division theorem)** *Let a and b be integers with b* > 0. *Then there are unique integers q and r such that*

This result should ...

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