1 Definitions and the Division Theorem In this set of notes, we look to develop a sense of division and divisibility in the integers. We begin by refreshing some definitions we may have seen before. Definition 1. Let a, be Z. We say that b divides a if there exists an integer & such that a = kb. The number b is called a divisor or factor of a, and the number a is called a multiple of b. Notationally, we write bla to denote that b divides a. We have already seen some results about division in our early work. For example, we showed the following things (possibly among others): . If b is even and bla, then a is even. . If a, be Z and a 2 2, then a does not divide one of b or b + 1. . If ab is even, then one of a or b is even. . If ab and bc, then alc. In addition, in constructing the rational numbers, we at least alluded to the idea that we could think about integer division with remainders, and then we left that alone. Time to revive it! We begin our foray into divisibility with the following theorem. Theorem 1 (Division Theorem). Let a, be Z with b / 0. Then there exist unique integers q, r e Z such that a = by +r and 0 Er 0. There are unique q, r c Z such that a = qb +r where 0 0. Then a