Unformatted text preview: u ≤ 2000 and 0 ≤ v ≤ 2000. 4. Suppose m, n ∈ Z . Prove that if m  n and n 6 = 0, then  m  ≤  n  . [4] 5. Suppose that e, f, g ∈ Z . Prove that if the linear Diophantine equation ex + fy = g has an [4] integer solution, then gcd( e, f )  g . 6. (a) Suppose that a, b, c ∈ P . Prove that if gcd( ac, b ) = 1, then gcd( a, b ) = 1. [4] (b) Suppose that a, b, d ∈ P . Prove that if gcd( a, b ) = 1 and d  a , then gcd( a, db ) = d . [4] 7. Suppose that x, y ∈ R with x > y > 0. [6] Prove by induction on n that ( x + y ) n ≥ x n +1y n +1 xy for all n ∈ P ....
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 Fall '08
 ANDREWCHILDS
 Math, Algebra, Diophantine equation, Linear Diophantine equation

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