M135F07A4

# M135F07A4 - r x is a constant r x = r ∈ Q(b Prove that r...

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MATH 135 Fall 2007 Assignment #4 Due: Thursday 11 October 2007, 8:20 a.m. Hand-In Problems 1. Which of the following linear Diophantine equations have solutions? In each case, explain brieﬂy why or why not. If there is a solution, determine the complete solution. (a) 28 x + 91 y = 40 (b) 2007 x - 897 y = 15 2. (a) Determine all non-negative integer solutions to the linear Diophantine equation 133 x + 315 y = 98000. (b) Determine all non-negative integer solutions to the linear Diophantine equation 133 x + 315 y = 98000 with the additional property that x 640 - 2 y . 3. Find the smallest positive integer x so that 141 x leaves a remainder of 21 when divided by 31. 4. Suppose a, b, c Z . Prove that gcd( a, c ) = gcd( b, c ) = 1 if and only if gcd( ab, c ) = 1. 5. Suppose a, b, n Z . Prove that if n 0, then gcd( an, bn ) = n · gcd( a, b ). 6. Suppose that f ( x ) is a polynomial of degree n with coeﬃcients from Q and suppose that g ( x ) = x - c for some c Q . (a) When f ( x ) is divided by g ( x ), we obtain a quotient q ( x ) and a remainder r ( x ). (See Assignment #3.) Explain why, in this case,

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Unformatted text preview: r ( x ) is a constant r ( x ) = r ∈ Q . (b) Prove that r = f ( c ). 7. Consider the system of equations a + b = 2 m 2 b + c = 6 m a + c = 2 Determine all real values of m for which a ≤ b ≤ c . (This problem is not directly related to the course material, but is included to keep your problem solving skills sharp.) Recommended Problems 1. Text, page 50, #42 2. Text, page 51, #44 3. Text, page 51, #48 4. Text, page 52, #75 5. Text, page 52, #79 ...continued 6. Let a , b and c be non-zero integers. Their greatest common divisor gcd( a, b, c ) is the largest positive integer that divides all of them. (a) If d = gcd( a, b, c ), prove that d is a common divisor of a and gcd( b, c ). (b) If f is a common divisor of a and gcd( b, c ), prove that f is a common divisor of a , b and c . (c) Prove that gcd( a, b, c ) = gcd( a, gcd( b, c ))....
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M135F07A4 - r x is a constant r x = r ∈ Q(b Prove that r...

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