8 - “An integer a is divisible by b”. “An integer b...

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Unformatted text preview: “An integer a is divisible by b”. “An integer b divides a”. What it means? Definition. An integer b divides a (b | a) if and only if there is an integer q such that a = qb, or equivalently ∃c, a = qb with U of D= Z. b is called a factor (divisor) of a, a is called a multiple of b. b a means NOT b|a (b does not divide a), i.e., ∀c, a = bc 1 Theorem. (Basic properties of division) (i) If a | b and b | c, then a | c. (ii) If a | b and a | c, then a | (xb + yc) for any x, y ∈ Z. In particular, a | (b + c) and a | (b − c). (iii) If a | b and b | a, then a = ±b. (iv) If a | b and b = 0, then |a| ≤ |b|. 3 Proof. (i). If b = qa and c = rb for some q, r ∈ Z, then c = rb = r(qa) = (rq)a. (ii). If b = qa and c = ra for some q, r ∈ Z, then xb + yc = x(ba) + y(ra) = (xb)a + (yr)a = (xb + yr)a. (iii) If b = qa and a = rb for some q, r ∈ Z, then a = rb = r(qa) = (rq)a, so that (rq − 1)a = 0. Case 1. a = 0. Then b = q · 0 = 0, and a = ±b. Case 2. a = 0. Then rq = 1, so that r = q = ±1. Hence a = ±b. (iv) Let b = qa for some q ∈ Z and b = 0. Then q = 0, so that |q| ≥ 1. Hence |b| = |q||a| ≥ |a|. 5 Definition. Let a and b are integers and b = 0. If a = qb + r and 0 ≤ r < |b|, then q is the quotient, r is remainder, when a is divided by a. An integer q is the quotient if and only if qb is the greatest multiple of b lesser than a. Examples. a = 14, b = 3, a = 4b + 2, q = 4, r = 2 a = 14, b = −3, a = −4b + 2, q = −4, r = 2 a = −14, b = 3, a = −5b + 1, q = −5, r = 1 a = −14, b = −3, a = 5b + 1, q = 5, r = 1 7 ...
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8 - “An integer a is divisible by b”. “An integer b...

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