{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

461hw4sol

# 461hw4sol - Math 461 F Homework 4 Solutions Spring 2011...

This preview shows pages 1–2. Sign up to view the full content.

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. Hence 3 2 is not rational.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

461hw4sol - Math 461 F Homework 4 Solutions Spring 2011...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online