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Unformatted text preview: Math 461 F Spring 2011 Homework 4 Solutions Drew Armstrong A.1. Euclid’s Lemma. Suppose that a divides bc for a,b,c ∈ Z with a and b coprime (i.e. they have no common factor except ± 1). Prove that a must divide c . (Hint: Since a and b are coprime, you may assume — without proof — that there exist x,y ∈ Z such that ax + by = 1.) Proof. Since a and b are coprime they have greatest common divisor 1. You may have seen in another class the fact that the greatest common divisor of ( a,b ) is always an integer linear combination of a and b . That is, there exist x,y ∈ Z such that ax + by = 1. Now multiply both sides of this equation by c to get axc + ( bc ) y = c. By assumption we have bc = ak for some k ∈ Z , hence axc + ( bc ) y = axc + aky = a ( xc + ky ) = c. In other words, a divides c . A.2. Prove that 3 √ 2 is not rational. Proof. Suppose for contradiction that 3 √ 2 is rational. Then we can write 3 √ 2 = a/b as a fraction in lowst terms (i.e. a,b ∈ Z with a,b coprime). Cubing both sides of this equation gives 2 = a 3 /b 3 , or a 3 = 2 b 3 . Since a 3 is even we may conclude that a is even (for if a were odd then a 3 would be odd), and we write a = 2 k for some k ∈ Z . But then we have 2 b 3 = a 3 = 8 a 3 , or b 3 = 4 a 3 , which implies that b 3 , and hence b , is even. We have found that a and b are both even, which contradicts the assumption that they are coprime. Hencecoprime....
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