# Greatest Common Divisors

May 14th, 2015
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Algebra Propositions Sheet Definition of Divides: An integer m divides an integer n, and we write m | n, if there exists an integer k so that n = km. Transitivity of Divisibility (TD)

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Greatest Common DivisorsAlgebra Propositions SheetDefinition of Divides:An integer m divides an integer n, and we write m | n, if there exists an integer k so that n = km. Transitivity of Divisibility (TD)Let a, b, c . If a | b and b | c, then a | c.Divisibility of Integer Combinations (DIC)Let a, b and c be integers. If a | b and a | c, then a | (bx + cy) for any x, y . Bounds By Divisibility (BBD)If a | b and b 0 then |a| |b|.Division Algorithm (DA)If a and b are integers, and b > 0, then there exist unique integers q and r such that a = qb + r, 0 r < b. Greatest Common Divisor (GCD)Let a and b be integers, not both zero. An integer d > 0 is the greatest common divisor of a and b, written gcd(a, b) = d, if and only if: (a) d | a and d | b (common), and (b) if c | a and c | b then c d (greatest). GCD With Remainders (GCD WR)If a and b are integers not both zero, and q and r such that a = qb + r, then gcd(a, b) = gcd(b, r). GCD Characterization Theorem (GCD CT)If d is a positive common divisor of the integers a and b, and there exist integers x and y so that ax+by= d, then d = gcd(a, b).Coprimeness and Divisibility

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