(a) Let a, b ? N, with b /= 0. Prove that there exist unique q, r ? N with r < b

Question

(a) Let a, b ? N, with b /= 0. Prove that there exist unique q, r ? N with r < b such that
a = qb + r. [Hint: to prove existence, fix b and use induction on a.]
(b) Extend this to the integers: let a, b ? Z, with b /= 0; prove that there exist unique q, r ? Z
with 0 ? r < |b| such that a = qb + r. [Hint: use (a) to prove existence, breaking up your
proof in various cases, according as a and b are in N or Z? .]
(c) Show that any ideal of Z is of the form nZ, for some n ? N. [Hint: if I /= {0}, let n be the smallest positive number in I . (Why does it exist?) Then use (b) to prove that I = nZ.]

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