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Unformatted text preview: LECTURE 29: DIVIDING ( 11.12) Nothing is particularly hard if you divide it into small jobs. Henry Ford (1863  1947) 1. Division facts we will use over and over Here are some definitions weve seen before. Definition 1. Let a,b Z . 1. We say that a divides b , or b is a multiple of a , or a  b , to mean that there exists some element x Z with b = ax . 2. An element p Z is prime if p 2 and the only positive divisors of p are 1 and p . 3. An element n Z is composite if n 2 and n is not prime. This is equivalent to saying that n = ab for some a,b Z , 1 &lt; a &lt; n and 1 &lt; b &lt; n . Theorem 2. Let a,b,c Z with a 6 = 0 . (i) If a  b then a  bc . (ii) If a  b and b  c then a  c . (iii) If a  b and a  c then a  ( bx + cy ) for any x,y Z . Proof. We do all of these directly. (i): Assume a  b . So there exists x Z with b = ax . Then bc = axc = a ( xc ). Since xc Z we have a  bc . (ii): Assume a  b and b  c . So there exists x Z with b = ax and y Z so that c = by . Thus c = by = axy = a ( xy ). Since xy Z we have a  c . (iii): Assume a  b and a  c . Then there exists u Z with b = au and v Z so that c = av . Let x,y Z . Then bx + cy = aux...
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
 Spring '11
 Smith
 Division

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