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

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

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