Greatest Common Divisor Operation

A Greatest Common Divisor Operation, \gcd \left({S}\right) where [math]\displaystyle{ S }[/math] is an integer set with one non-zero element, is a integer set operation that produces the largest positive integer that divides the numbers without a remainder.

• See: Euclid's Algorithm.

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References

2013

• http://en.wikipedia.org/wiki/Greatest_common_divisor
  • QUOTE: In mathematics, the greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more integers (at least one of which is not zero), is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.

    This notion can be extended to polynomials, see Polynomial greatest common divisor, or to rational numbers (with integer quotients).

• http://www.proofwiki.org/wiki/Definition:Greatest_Common_Divisor/Integers/General_Definition
  • Let [math]\displaystyle{ S = \left\{{a_1, a_2, \ldots, a_n}\right\} \subseteq \Z }[/math] such that [math]\displaystyle{ \exists x \in S: x \ne 0 }[/math] (that is, at least one element of [math]\displaystyle{ S }[/math] is non-zero).

    Then: : [math]\displaystyle{ \gcd \left({S}\right) = \gcd \left\{{a_1, a_2, \ldots, a_n}\right\} }[/math] is defined as the largest [math]\displaystyle{ d \in \Z_{\gt 0} }[/math] such that [math]\displaystyle{ \forall x \in S: d \mathop \backslash x }[/math].

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