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

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McCombs Math 381 Greatest Common Divisor 1 Formal Definition: Given , a b Z , with 0 a or 0 b . The greatest common divisor of a and b , written ( ) gcd , a b , is the largest integer that divides both a and b . In other words, ( ) gcd , a b d = means that | d a and | d b , and if | c a and | c b , then c d < . Special Case: If ( ) gcd , 1 a b = , we say that a and b are relatively prime (coprime) . Other Important Results: 1. Given , , a b w Z , the equation ax by d + = has integer solutions , x y if and only if ( ) gcd , | a b w . 2. Given , , a b c positive integers such that ( ) gcd , 1 a b = . If ( ) | a bc , then | a c . 3. Given 1 2 3 , , ,..., n a a a a Z and any prime number p . If ( ) 1 2 3 | ... n p a a a a , then | k p a for some 1 k n . 4. Given integers 3 1 2 1 2 3 ... a a a a n n a p p p p = and 3 1 2 1 2 3 ... b b b b n n b p p p p = , where each exponent is a nonnegative integer, and 1 2 3 , , ,..., n p p p p are prime. (i) ( ) ( ) ( ) ( ) ( ) , , , , min min min min 3 4 1 1 2 2 1 2 3 gcd , ... a b a b a b a b n n n a b p p p p = (ii) ( ) ( ) ( ) ( ) ( ) , , , , max max max max 3 4 1 1 2 2 1 2 3 lcm , ... a b a b a b a b n n n a b p p p p = (iii) ( ) ( ) lcm , gcd , a b a b a b = Crucial Theorem: If ( ) gcd , a b d = , there exist integers , x y such that ax by d + = .

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

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