MATH 245 CH4 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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DISCRETE MATHEMATICS W W L CHEN c ± W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 4 DIVISION AND FACTORIZATION 4.1. Division Definition. Suppose that a,b Z and a 6 = 0. Then we say that a divides b , denoted by a | b , if there exists c Z such that b = ac . In this case, we also say that a is a divisor of b , or b is a multiple of a . Example 4.1.1. For every a Z \ { 0 } , a | a and a | - a . Example 4.1.2. For every a Z , 1 | a and - 1 | a . Example 4.1.3. If a | b and b | c , then a | c . To see this, note that if a | b and b | c , then there exist m,n Z such that b = am and c = bn , so that c = amn . Clearly mn Z . Example 4.1.4. If a | b and a | c , then for every x,y Z , a | ( bx + cy ). To see this, note that if a | b and a | c , then there exist m,n Z such that b = am and c = an , so that bx + cy = amx + any = a ( mx + ny ). Clearly mx + ny Z . PROPOSITION 4A. Suppose that a N and b Z . Then there exist unique q,r Z such that b = aq + r and 0 r < a . Proof. We shall first of all show the existence of such numbers q,r Z . Consider the set S = { b - as 0 : s Z } . Then it is easy to see that S is a non-empty subset of N ∪ { 0 } . It follows from the Well-ordering principle that S has a smallest element. Let r be the smallest element of S , and let q Z such that b - aq = r . Clearly r 0, so it remains to show that r < a . Suppose on the contrary that r a . Then b - a ( q + 1) = ( b - aq ) - a = r - a 0, so that b - a ( q + 1) S . Clearly b - a ( q + 1) < r , contradicting that r is the smallest element of S . Chapter 4 : Division and Factorization page 1 of 6
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Discrete Mathematics c ± W W L Chen, 1982, 2008 Next we show that such numbers q,r Z are unique. Suppose that b = aq 1 + r 1 = aq 2 + r 2 with 0 r 1 < a and 0 r 2 < a . Then a | q 1 - q 2 | = | r 2 - r 1 | < a . Since | q 1 - q 2 | ∈ N ∪ { 0 } , we must have | q 1 - q 2 | = 0, so that q 1 = q 2 and so r 1 = r 2 also. ± Definition. Suppose that a N and a > 1. Then we say that a is prime if it has exactly two positive divisors, namely 1 and a . We also say that a is composite if it is not prime. Remark. Note that 1 is neither prime nor composite. There is a good reason for not including 1 as a prime. See the remark following Proposition 4D.
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MATH 245 CH4 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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