# 30 - LECTURE 30 GREATEST COMMON DIVISORS(11.3-4 The...

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

LECTURE 30: GREATEST COMMON DIVISORS ( § 11.3-4) The greatest use of life is to spend it for something that will outlast it. William James (1842 - 1910) 1. The Division Algorithm Examples From last time: Theorem 1. Given integers n,d Z (called the numerator and denominator), with d 6 = 0 , there exist unique integers q,r Z (called the quotient and remainder) such that n = qd + r and 0 r < | d | . In fact, we can take q = b n/d c and r = n - qd . It turns out quotients are unimportant. It is remainders which are important. This has to do with congruence equations. Notice that n = qd + r says that n - r = qd so d | n - r , hence n r (mod d ). So numerators are equivalent to their remainders, modulo the denominator. Example 2. If our denominator is d = 4 what kinds of remainders can we get? Only 0 , 1 , 2 , 3. Try some examples. What is the remainder when n = 16? What if n = 10393? What if n = - 1? (The remainder here is 3.) Every integer is of the form 4 k + 0, 4 k + 1, 4 k + 2, or 4 k + 3, depending only on the remainder. Example 3.

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.

## This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.

### Page1 / 2

30 - LECTURE 30 GREATEST COMMON DIVISORS(11.3-4 The...

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

View Full Document
Ask a homework question - tutors are online