This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: LECTURE 29: DIVIDING ( 11.1-2) 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...
View Full Document
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