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The greatest common divisor of ''a'' and ''b'' is written as gcd(''a'', ''b''), or sometimes simply as (''a'', ''b''). For example, gcd(12, 18) = 6, gcd(−4, 14) = 2 and gcd(5, 0) = 5. Two numbers are called '' Coprime '' or ''relatively prime'' if their greatest common divisor equals 1. For example, 9 and 28 are relatively prime.

The greatest common divisor is useful for reducing Vulgar Fraction s to be In Lowest Terms . Consider for instance
:{42 \over 56}={3 \cdot 14 \over 4 \cdot 14}={3 \over 4}
where we cancelled 14, the greatest common divisor of 42 and 56.


CALCULATING THE GCD

Greatest common divisors can in principle be computed by determining the Prime Factorizations of the two numbers and comparing factors, as in the following example: to compute gcd(18,84), we find the prime factorizations 18 = 2·32 and 84 = 22·3·7 and notice that the "overlap" of the two expressions is 2·3; so gcd(18,84) = 6. In practice, this method is only feasible for very small numbers; computing prime factorizations in general takes far too long.

A much more efficient method is the Euclidean Algorithm : divide 84 by 18 to get a quotient of 4 and a remainder of 12. Then divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then divide 12 by 6 to get a remainder of 0, which means that 6 is the gcd.


PROPERTIES

  • Every common divisor of ''a'' and ''b'' is a divisor of gcd(''a'', ''b'').


  • gcd(''a'', ''b''), where ''a'' and ''b'' are not both zero, may be defined alternatively and equivalently as the smallest positive integer ''d'' which can be written in the form ''d'' = ''a''·''p'' + ''b''·''q'' where ''p'' and ''q'' are integers. This expression is called Bézout's Identity . Numbers ''p'' and ''q'' like this can be computed with the Extended Euclidean Algorithm .


  • If ''a'' divides the product ''b''·''c'', and gcd(''a'', ''b'') = ''d'', then ''a''/''d'' divides ''c''.


  • If ''m'' is any integer, then gcd(''m''·''a'', ''m''·''b'') = ''m''·gcd(''a'', ''b'') and gcd(''a'' + ''m''·''b'', ''b'') = gcd(''a'', ''b''). If ''m'' is a nonzero common divisor of ''a'' and ''b'', then gcd(''a''/''m'', ''b''/''m'') = gcd(''a'', ''b'')/''m''.


  • The gcd is a Multiplicative Function in the following sense: if ''a''1 and ''a''2 are relatively prime, then gcd(''a''1·''a''2, ''b'') = gcd(''a''1, ''b'')·gcd(''a''2, ''b'').


  • The gcd of three numbers can be computed as gcd(''a'', ''b'', ''c'') = gcd(gcd(''a'', ''b''), ''c'') = gcd(''a'', gcd(''b'', ''c'')). Thus the gcd is an Associative operation.


  • gcd(''a'', ''b'') is closely related to the Least Common Multiple lcm(''a'', ''b''): we have

    • ::gcd(''a'', ''b'')·lcm(''a'', ''b'') = ''a''·''b''.

:This formula is often used to compute least common multiples: one first computes the gcd with Euclid's algorithm and then divides the product of the given numbers by their gcd. The following versions of Distributivity hold true:
::gcd(''a'', lcm(''b'', ''c'')) = lcm(gcd(''a'', ''b''), gcd(''a'', ''c''))
::lcm(''a'', gcd(''b'', ''c'')) = gcd(lcm(''a'', ''b''), lcm(''a'', ''c'')).

  • It is useful to define gcd(0, 0) = 0 and lcm(0, 0) = 0 because then the Natural Number s become a Complete Distributive Lattice with gcd as meet and lcm as join operation. This extension of the definition is also compatible with the generalization for commutative rings given below.


  • In a Cartesian Coordinate System , gcd(''a'', ''b'') can be interpreted as the number of points with integral coordinates on the Straight Line joining the points (0, 0) and (''a'', ''b''), excluding (0, 0).



THE GCD IN COMMUTATIVE RINGS

The greatest common divisor can more generally be defined for elements of an arbitrary Commutative Ring .

If ''R'' is a commutative ring, and ''a'' and ''b'' are in ''R'', then an element of ''d'' of ''R'' is called a ''common divisor'' of ''a'' and ''b'' if it divides both ''a'' and ''b'' (that is, if there are elements ''x'' and ''y'' in ''R'' such that ''d''·''x'' = ''a'' and ''d''·''y'' = ''b'').
If ''d'' is a common divisor of ''a'' and ''b'', and every common divisor of ''a'' and ''b'' divides ''d'', then ''d'' is called a ''greatest common divisor'' of ''a'' and ''b''.

Note that with this definition, two elements ''a'' and ''b'' may very well have several greatest common divisors, or none at all. But if ''R'' is an Integral Domain then any two gcd's of ''a'' and ''b'' must be Associate Elements . Also, if ''R'' is a Unique Factorization Domain , then any two elements have a gcd.
If ''R'' is a Euclidean Domain then a form of the Euclidean algorithm can be used to compute greatest common divisors.

The following is an example of an integral domain with two elements that do not have a gcd:
:R = \mathbb{Z} {Link without Title} ,\quad a = 4 = 2\cdot 2 = (1+\sqrt{-3})(1-\sqrt{-3}),\quad b = (1+\sqrt{-3})\cdot 2
The elements 1+\sqrt{-3} and 2 are two "maximal common divisors" (i.e. any common divisor which is a multiple of 2 is associated to 2, the same holds for 1+\sqrt{-3}), but they are not associated, so there is no greatest common divisor of ''a'' and ''b''.

Corresponding to the Bezout property we may, in any commutative ring, consider the collection of elements of the form p a + q b, where ''p'' and ''q'' range over the ring. This is the Ideal generated by ''a'' and ''b'', and is denoted simply (a,b). In a ring all of whose ideals are principal (a Principal Ideal Domain or PID), this ideal
will be identical with the set of multiples of some ring element ''d''; then this ''d'' is a greatest common divisor of ''a'' and ''b''. But the ideal (a,b) can be useful even when there is no greatest common divisor of ''a'' and ''b''. (Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem , although he envisioned it as the set of multiples of some hypothetical, or ''ideal'', ring element ''d'', whence the ring-theoretic term.)


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