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

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“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

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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 = ±

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