M135F08A4 - MATH 135 Assignment#4 Hand-In Problems Fall...

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MATH 135 Fall 2008 Assignment #4 Due: Wednesday 15 October 2008, 8:20 a.m. Hand-In Problems 1. For each pair a and b , state the quotient and remainder when a is divided by b . (a) a = 261, b = 12 (b) a = - 261, b = 12 (c) a = 261, b = - 12 (d) a = - 261, b = - 12 2. A quadratic equation ax 2 + bx + c = 0 (with a, b, c 6 = 0) has real roots. Prove by contradiction that a, b, c in that order cannot be consecutive terms of a geometric sequence. 3. (a) Determine gcd(1989 , 251). (b) Determine integers x and y such that 1989 x + 251 y = gcd(1989 , 251). 4. (a) Determine gcd(686 , 511). (b) Determine integers x and y such that 686 x + 511 y = gcd(686 , 511). 5. If ac | bc and c 6 = 0, prove that a | b . 6. Suppose that m, n P . Prove that if m 2 6 | n 2 , then m 6 | n . (Hint: Use the contrapositive.) 7. Suppose a, b, n Z . Prove that if n > 0, then gcd( an, bn ) = n · gcd( a, b ). 8. (a) Determine the quotient and remainder when f ( x ) = - x 5 + x 4 + 11 x 3 - 18 x 2 + 20 x + 4 is divided by
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This note was uploaded on 09/04/2009 for the course MATH 135 taught by Professor Andrewchilds during the Fall '08 term at Waterloo.

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M135F08A4 - MATH 135 Assignment#4 Hand-In Problems Fall...

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