74-hw3 - MATH 74 HOMEWORK 3: DUE MONDAY 2/12 1. If a and b...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 74 HOMEWORK 3: DUE MONDAY 2/12 1. If a and b are natural numbers, and b > a , and a 6 | b , then there are q N and 0 < r < a satisfying b = qa + r. In class we proved gcd( a, b ) = gcd( a, r ) by iterating a subtration rule gcd( a, b ) = gcd( a, b - a ) repeatedly. In this exercise you will give a different, more direct proof. 1(a). Let a, b, q, r be as above. Suppose you have a natural number x that is in D ( a, b ). Show, making explicit references to the definitions of the relevant things, that x is also an element of D ( a, r ). 1(b). Give a similarly explicit proof that if x is in D ( a, r ) then x is in D ( a, b ). Remark. The D ( , ) notation is explained in the lecture outlines, if you missed it. Together, the statements 1(a) and 1(b) show that D ( a, r ) = D ( a, b ) , and then gcd( a, r ) = gcd( a, b ) follows from the definition of gcd . Remark. In the previous exercise, you showed that two pairs of numbers had the same gcd by showing that they had the same set of common divisors. This raises
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This homework help was uploaded on 04/02/2008 for the course MATH 74 taught by Professor Courtney during the Fall '07 term at University of California, Berkeley.

Ask a homework question - tutors are online