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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 non-zero 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 non-zero 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).