MST10040_2013_Lec16 - Greatest Common Divisor Denition Let...

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Greatest Common Divisor Definition Let b and c be any two integers. A positive integer d is called the greatest common divisor (gcd) of b and c if d divides both b and c and is the largest integer with this property. We write gcd ( b , c ) for the gcd of b and c . Lecture 16 MST10040 November 1, 2013 1 / 11
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Greatest Common Divisor Theorem Let b and c be any two non-zero integers and let d be their gcd. Then there exist integers s and t with d = sb + tc . Proof: Omitted. Lecture 16 MST10040 November 1, 2013 2 / 11
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Greatest Common Divisor Corollary Let b and c be any two non-zero integers. Then any common divisor of b and c divides their gcd. Lecture 16 MST10040 November 1, 2013 3 / 11
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Greatest Common Divisor Fact For any two integers b and c we have gcd ( b , c ) = gcd ( b + kc , c ) for any integer k. Lecture 16 MST10040 November 1, 2013 4 / 11
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Greatest Common Divisor Proof: If a is a common factor of b and c then a divides both b and c . Hence a also divides kc for any integer k and so a divides b + kc . Thus a is a common factor of b + kc and c . Similarly, if a is a common factor of b + kc and c then a divides both b + kc and c . Hence a also divides ( b + kc ) - kc = b . Thus a is a common factor of b and c . Hence the set of common factors of b and c is the same as the set of common factors of b + kc and c . Now, since the two sets are the same, their largest elements are the same. The largest element of the first set is gcd ( b , c ) and the largest element of the second set is gcd ( b + kc , c ) . Hence gcd ( b , c ) = gcd ( b + kc , c ) . Lecture 16 MST10040 November 1, 2013 5 / 11
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Greatest Common Divisor Proof: If a is a common factor of b and c then a divides both b and c . Hence a also divides kc for any integer k and so a divides b + kc . Thus a is a common factor of b + kc and c . Similarly, if a is a common factor of b + kc and c then a divides both b + kc and c . Hence a also divides ( b + kc ) - kc = b . Thus a is a common factor of b and c .
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  • Fall '13
  • Natural number, Greatest common divisor, Euclidean algorithm, gcd

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