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Unformatted text preview: McCombs Math 81 Working with gcd( a , b ) 1 Important Theorems: Theorem: Given , a b ! Z , with a and b not both 0. If ( ) gcd , d a b = , then there exist , x y ! Z such that d ax by = + . Theorem: Given , a b ! Z , with a and b not both 0. If ( ) gcd , d a b = , then d is the smallest positive integer that can be expressed as the linear combination d ax by = + , where , x y ! Z . Theorem: Integers a and b are relatively prime if and only if there exist , x y ! Z such that 1 ax by + = . Proof Examples: 1. Given , , a b w ! Z . The equation ax by w + = has integer-valued solutions , x y if and only if ( ) gcd , a b w . Part (i): Assume the equation ax by w + = has integer-valued solutions , x y . Let ( ) gcd , d a b = . We know that a kd = and B md = for some integers m and n . ( ) ( ) ax by w kd x md y w + = ! + = So ( ) w d kx my = + . Since k , x , m , and y are all integers, d w . Thus, ( ) gcd , a b w ....
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- Summer '11