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31 - Common Introduction Divisors Discrete Mathematics...

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Introduction Common Divisors Discrete Mathematics Andrei Bulatov

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Discrete Mathematics – Common Divisors 31-2 Previous Lecture Representation of numbers Prime and composite numbers
Discrete Mathematics – Common Divisors 31-3 The Greatest Common Divisor For integers a and b, a positive integer c is said to be a common divisor of a and b if c | a and c | b Let a, b be integers such that a 0 or b 0. Then a positive integer c is called the greatest common divisor of a, b if (a) c | a and c | b (that is c is a common divisor of a, b) (b) for any common divisor d of a and b, we have d | c The greatest common divisor of a and b is denoted by gcd(a,b)

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Discrete Mathematics – Common Divisors 31-4 The Greatest Common Divisor (cntd) Theorem For any positive integers a and b, there is a uniquepositive integer c such that c is the greatest common divisor of a and b First try: Take the largest common divisor, in the sense of usual order Does not work: Why every other common divisor divides it? 4 1 2 3 5 11 7 10 6 9 12 8 a b gcd(a,b)
Discrete Mathematics – Common Divisors 31-5 The Greatest Common Divisor (cntd) Proof . Given a, b, let S = { as + bt | s,t Z , as + bt > 0 }. Since S ≠ ∅ , it has a least element c. We show that c = gcd(a,b) We have c = ax + by for some integers x and y. If d | a and d | b, then d | ax + by = c. If c | a, we can use the division algorithm to find a = qc + r, where q,r are integers and 0 < r < c.

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