lecture20 - MA1100 Lecture 20 Number Theory Greatest Common...

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MA1100 Lecture 20 Number Theory Greatest Common Divisor Division Algorithm Euclidean Algorithm Linear combination Chartrand: 11.2, 11.3, 11.4
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Summary •fandgin ject ive • f and g surjective •g o f injective •g o f surjective •fb i ject ion f -1 is a bijection • f(a) = b •fb i ject ion f -1 o f = and f o f -1 = •g o f = I •g o f = I and f o g= I •(g o f)
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Common divisor Recall 1 Recall 2 Let a, b be integers and d a nonzero integer. Let n be an integer and m a non-zero integer. We say m divides n and write m | n if there exists an integer q such that n = mq . We also say m is a divisor of n. We say d is a common divisor (factor) of a and b if d | a and d | b . 3 Lecture 20 Chartrand p.250
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Common divisor Example Let a = 48 and b = 84. Divisors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Divisors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 Common divisors of 48 and 84: The greatest common divisor of 48 and 84 is 4 Lecture 20
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Greatest common divisor Definition Notation The largest integer that divides both a and b is called the greatest common divisor of a and b. gcd(a, b) Let a, b be integers, not both 0. Example gcd(48, 84) = 12 Remark gcd(0, 0) = ? 5 Lecture 20 Chartrand p.250
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Basic properties of gcd 1.gcd(a, b) > 0 2.gcd(a, a) = 3.gcd(a, 0) = 4.gcd(a, b) = gcd(-a, b) = gcd(a, -b) = gcd(-a, -b) Let a and b be integers with a 0. 6 Lecture 20
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Definition of gcd Rephrasing the definition Let a, b be integers, not both 0, and d œ N . d gcd(a, b) if negation working definition d = gcd(a, b) if d | a and d | b and • For all k œ N ,i f k | a and k | b , then k § d 7
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Finding gcd Example 1. List and compare 2. Prime factorization 3. Euclidean Algorithm Methods of finding GCD: 48 = 2 ä 2 ä 2 ä 2 ä 3 84 = 2 ä 2 ä 3 ä 7 8 Lecture 20
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lecture20 - MA1100 Lecture 20 Number Theory Greatest Common...

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