Unformatted text preview: (a) Show that there is no solution to the pair of diophantine equations 2 x + 7 y + z = 45 3 x + 8 y + 4 z = 21 . (b) Find all solutions to the pair of diophantine equations 20 x + 12 y + 15 z = 85 18 x + 20 y + 8 z = 110 . 5: Let a , b and c be nonzero integers. The greatest common divisor d = gcd( a,b,c ) is deﬁned to be the largest positive integer d such that d ± ± a , d ± ± b and d ± ± c . (a) Show that gcd( a,b,c ) = gcd ( gcd( a,b ) ,c ) . (b) Show that for any integer e , the linear diophantine ax + by + cz = e has a solution if and only if gcd( a,b,c ) ± ± e . (c) Find all solutions to the linear diophantine equation 42 x + 70 y + 105 z = 63....
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This note was uploaded on 07/25/2011 for the course MATH 135 taught by Professor Andrewchilds during the Spring '08 term at Waterloo.
 Spring '08
 ANDREWCHILDS
 Algebra, Equations

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