Jis aiso of the form 31+2, but not conver 3.use the division algorithm to establish the following: (a) the square of any integer is either of the form 3k or 3k +1 (bdthe cube of any integer has one of the forms: 9k, 9k + 1, or 9k +8.
2.2 the division algorithm we have been exposed to relationships between intege for several pags and, as single divisibility property has been derived. it is time to remedy this yet, not a situation. one theorem, the division algorithm, acts as the foundation stone upon which our whole development rests. the result is familiar to most of us: roughly, it integer asserts that an the remainder is smaller than is b. the exact statement of this fact is theorem 2.1. a can be "divided" by a positive integer b in such a way that theorem 2.1 division algorithm. given integers a and b, with b> 0, there exist unique integers q and r satisfying qb+r 0sr 1, we have |a | b> al, and so a - (-la |)b= a +la lbz a+la |0 la 1, then, a - xb lies in s. this paves the way for an application of the well-ordering principle (chapter 1), from which we infer that the set s contains integer q satisfying for the choice x = a smallest integer; call it r. by the definition of s, there exists an 0r raqb we argue that r b and a - (q+1)b (a -qb)- b r-b20 the implication is that the integer a - (q + 1)b has the proper form to belong set s. buta- the smallest member of s. hence, r