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Unformatted text preview: CS 2800: Discrete Math Sept. 21, 2011 Complimentary Definitions Lecturer: John Hopcroft Scribe: June Andrews Number Theory Greatest common divisor (GCD) The greatest common divisor of two or more nonzero integers is the largest positive integer that divides each of the integers evenly (i.e., without a remainder). We use gcd( a 1 ,a 2 ,...,a n ), where n 2, to denote the greatest common divisor of nonzero integers a 1 ,a 2 ,...,a n . In general, gcd( a 1 ,a 2 ,...,a n ) can be computed by finding the prime factorizations of a 1 ,a 2 ,...,a n . In the special case where n = 2, the GCD can also be computed by using the Euclidean algorithm. Examples. The greatest common factor of 8 and 12 is 4, and the greatest common factor of 12 , 20 , and 30 is 2. For any integer a , gcd( a, 0) = a and gcd( a, 1) = 1. Relatively Prime Two integers a and b are said to be coprime or relatively prime if they have no common positive divisors other than 1. Equivalently, a and b are relatively prime if their greatest common divisor is 1. One writesare relatively prime if their greatest common divisor is 1....
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This note was uploaded on 11/11/2011 for the course CS 2800 at Cornell University (Engineering School).
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