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Unformatted text preview: Serge Ballif MATH 535 Homework 1 August 31, 2007 1) (a) Find integers x and y such that 4 x + 11 y = 1 . (b) Explain how all pairs ( x,y ) of integers 4 x + 11 y = 1 are related to the pair you found. (a) The ordered pair ( x ,y ) = (3 ,- 1) satisfies the equation. (b) The line 4 x +11 y = 1 (or y =- 4 11 x + 1 11 ) goes through the point (3 ,- 1), but as x varies from one integer to the next, we see that y is increasing (or decreasing) by 4 11 . Since 4 and 11 are relatively prime, the only way for the y-value to be an integer at the same time as the x-value is when the x-value is of the form 3+11 m for some integer m . Thus all pairs of integer solutions of 4 x + 11 y = 1 are of the form (3 + 11 m,- 1- 4 m ). 2) Let P ( X ) and Q ( X ) Q [ X ] be polynomials. Prove that if F ( X ) is a gcd of P ( X ) and Q ( X ) in C [ X ] , then for some non-zero C , F ( X ) Q [ X ] . F ( X ) is a gcd of P ( X ) and Q ( X ) all divisors of both P ( X ) and Q ( X ) divide F ( X ). Let G ( X ) Q [ X ] be a gcd of P ( X ) and Q ( X ) in Q [ X ]. Since Q [ X ] C [ X ], we know that G ( X ) divides F ( X ). However, F ( X ) divides both P ( X ) and Q ( X ), so it divides any sum of multiples of...
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