Further concepts and computations.   7) Much like we defined addition modulo n in Zn, we can also define multiplication modulo n. For example in Z5, 4+n 3=(4+3)mod5=7mod5=2,wecanalsodefine multiplication modulo n. For example in Z5,3 .5 4=3×4mod5=12mod5=2because 3 × 4 = 12 = 2(5) + 2. Show that the subset {1, 3, 5, 7} of Z8 with multiplication modulo 8, .8 is a group, and give the table for this group.

22. Evaluation of proofs See the instructions for Exercise (19) on page 100 from Section 3.1. (a) Proposition. For all integers a and b, if (a + 2b) = 0 (mod 3), then (2а + b) %3D 0 (mod 3). Proof. We assume a, b e Z and (a + 2b) = 0 (mod 3). This means that 3 divides a + 2b and, hence, there exists an integer m such that a + 2b — Зт. Hence, a %3 3m – 2b. For (2a +b) = 0 (mod 3), there exists an integer x such that 2a + b = 3x. Hence, 2(Зт — 2b) + b %3 3x бт — 3Ь — 3х 3(2m – b) = 3x 2m — b 3 х. Since (2m – b) is an integer, this proves that 3 divides (2a + b) and hence, (2a + b) = 0 (mod 3). (b) Proposition. For each integer m, 5 divides (m³ – m). Proof. Let m e Z. We will prove that 5 divides (m³ – m) by proving that (m – m) = 0 (mod 5). We will use cases. For the first case, if m = 0 (mod 5), then m³ = 0 (mod 5) and, hence, (m³ – m) = 0 (mod 5). For the second case, if m = 1 (mod 5), then m³ = 1 (mod 5) and, hence, (m³ – m) = (1 – 1) (mod 5), which means that (m³ – m) 0 (mod 5). For…

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(a) Find the order of a modulo 7 for a = 1, 2, 3, 4, 5, 6.