31a-Common-Divisors

31a-Common-Divisors - Common Introduction Divisors Discrete...

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Common Divisors Discrete Mathematics
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Discrete Mathematics – Common Divisors 31-2 Previous Lecture Representation of numbers Prime and composite numbers Common divisors The greatest common divisor
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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 Euclidean Algorithm: Small Example To warm up, let us find the greatest common divisor of 287 and 91 287 = 91 3 + 14 Note that any common divisor of 287 and 91 is also a divisor of 14 = 287 – 91 3. Conversely, every common divisor of 91 and 14 is also a divisor of 287 = 91 3 + 14. Thus gcd(287,91) = gcd(91,14). Next 91 = 14 6 + 7. By the same argument gcd(91,14) = gcd(14,7). Finally, since 7 | 14, gcd(14,7) = 7. Thus, gcd(287,91) = 7.
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Discrete Mathematics – Common Divisors 31-5 Euclidean Algorithm: Key Property Lemma . Let a = bq + r, where a, b, q, and r are integers. Then gcd(a,b) = gcd(b,r) Proof Let d be a common divisor of a and b. Then d also divides r = a – bq. Thus, d is a common divisor of b and r. Now, let d be a common divisor of b and r. Then d also divides a = bq + r. Therefore the pairs a,b and b,r have the same common divisors. Hence, gcd(a,b) = gcd(b,r).
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Discrete Mathematics – Common Divisors 31-6 Euclidean Algorithm: The Algorithm Let a and b be positive integers with a b. Set and Successively apply the division algorithm until the remainder is 0 Eventually, the remainder is zero, because the sequence of remainders cannot contain more than a elements.
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This note was uploaded on 12/12/2010 for the course MACM 201 taught by Professor Marnimishna during the Fall '09 term at Simon Fraser.

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31a-Common-Divisors - Common Introduction Divisors Discrete...

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