gcdkey2 - McCombs Math 381 Greatest Common Divisor Given a,...

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McCombs Math 381 Greatest Common Divisor 1 Formal Definition: Given , ab Z , with 0 a or 0 b . The greatest common divisor of a a n d b , written ( ) gcd , , is the largest integer that divides both a a n d b . In other words, ( ) gcd , d = means that | da and | db , and i f | ca and | cb , then cd < . Special Case: If () gcd , 1 = , we say that a and b are relatively prime (coprime) . Other Important Results: 1. Given ,, abw Z , the equation ax by d + = has integer solutions , x y if and only if gcd , | ab w . 2. Given abc positive integers such that ( ) gcd , 1 = . If | c , then | ac . 3. Given 123 , , ,..., n aa a a Z and any prime number p . If | ... n p a ⋅⋅⋅ , then | k p a for some 1 kn . 4. Given integers 3 12 3 ... a aa a n n ap p p p =⋅ and 3 3 ... b bb b n n bp p p p , where each exponent is a nonnegative integer, and , , ,..., n p pp p are prime. (i) ( ) ( ) ( ) , , min min min min 34 11 2 2 3 gcd , ... a b nn n p p p p (ii) ( ) ( ) ( ) , , max max max max 3 lcm , ... n p p p p (iii) ( ) ( ) lcm , gcd , ⋅= Crucial Theorem: If ( ) gcd , d = , there exist integers , x y such that ax by d += . Integers a and b are relatively prime if and only if there exist integers , xy such that 1 ax by + = .
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This note was uploaded on 06/16/2011 for the course MATH 381 taught by Professor Na during the Summer '11 term at University of North Carolina School of the Arts.

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gcdkey2 - McCombs Math 381 Greatest Common Divisor Given a,...

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