230hw3 - k . Problem 3. Use the Extended Euclidean...

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Math 230 E Fall 2011 Homework 3 Drew Armstrong Problem 1. To prove the following properties of Z you may assume axioms (A1) through (D) from the handout. (a) Prove that 0 a = 0 for all a Z . (b) Recall that - a is the unique integer such that a +( - a ) = 0. Prove that for all a,b Z we have ( - a )( - b ) = ab . (Hint: Show that ( - a )( - b ) + a ( - b ) = ab + a ( - b ). You will need the result from part (a).) Problem 2. The Division Algorithm 2.12 says that for all a,b Z with b > 0 there exist unique q,r Z such that a = qb + r and 0 r < b . Use this to prove the following Theorem: For all a,b Z with b > 0 there exists a unique k Z such that k a b < k + 1 . Note: You must prove both the existence and the uniqueness of
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Unformatted text preview: k . Problem 3. Use the Extended Euclidean Algorithm 2.25 to compute d = gcd(3953 , 1829) and then find two different pairs x,y ∈ Z such that 3953 x + 1829 y = d . Problem 4. We say a,b ∈ Z are coprime if gcd( a,b ) = 1. For all a,b,c ∈ Z prove that ab is coprime to c if and only if a and b are both coprime to c . (Hint: Use the GCD Characterization Theorem 2.24.) Problem 5. Prove that there is no perfect square of the form 4 k + 3....
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