Greatest common divisor

highest common factor

The largest of the common divisors of a set of integers or, in particular, of natural numbers $a_1,\ldots,a_n$. The greatest common divisor of a set of numbers not all of which are zero always exists. The greatest common divisor of $a_1,\ldots,a_n$ is usually denoted by $(a_1,\ldots,a_n)$.

Properties of the greatest common divisor are:

1) The greatest common divisor of $a_1,\ldots,a_n$ is divisible by any common divisor of these numbers.

2) $(a_1,\ldots,a_n,a_{n+1}) = ((a_1,\ldots,a_n),a_{n+1})$.

3) If $a_1,\ldots,a_n$ are expressed as $$ a_1 = p_1^{\alpha_1} \cdots p_s^{\alpha_s}\ ,\ \ \ \ldots\ a_n = p_1^{\nu_1} \cdots p_s^{\nu_s} $$ where $p_1,\ldots,p_s$ are distinct primes, $\alpha_i \ge 0\,\,\ldots\,\,\nu_i \ge 0$, $i=1,\ldots,s$ and $\delta_i = \min\{\alpha_i,\ldots,\nu_i\}$, then $$ (a_1,\ldots,a_n) = p_1^{\delta_1} \cdots p_s^{\delta_s} \ . $$

The greatest common divisor of two natural numbers can be determined by the Euclidean algorithm. The number of steps necessary to do this is bounded from above by five times the number of digits in the smaller of the two numbers (in decimal notation).

A greatest common divisor of two polynomials over a given field is a polynomial of greatest degree that divides both polynomials. In this case again, a greatest common divisor is divisible by any other common divisor of the polynomials: cf Factorization of polynomials. The Euclidean algorithm may be used in this setting also to obtain the highest common factor, with a number of steps bounded by the smaller of the degrees of the polynomials: cf Euclidean ring

A greatest common divisor of elements of an integral domain is defined as a common divisor of these elements that is divisible by any other common divisor. In general, a greatest common divisor of two elements of an integral domain need not exist (cf Divisibility in rings, but if one exists, it is unique up to multiplication by an invertible element. The greatest common divisor of two ideals $\mathfrak{a}$ and $\mathfrak{b}$ in a ring is the ideal $(\mathfrak{a},\mathfrak{b})$ generated by the union of the sets$\mathfrak{a}$ and $\mathfrak{b}$ (see Factorial ring).

References

[1]	I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
[2]	A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian)
[3]	A.I. Markushevich, "Division with remainder in arithmetic and algebra" , Moscow-Leningrad (1949) (In Russian)
[4]	R. Faure, A. Kaufman, M. Denis-Papin, "Mathématique nouvelle" , Dunod (1964)
[5]	S. Lang, "Algebra" , Addison-Wesley (1974)
[6]	K. Ireland, M. Rosen, "A classical introduction to modern number theory" , Springer (1982)

Comments

More generally, if $R$ is a domain, a greatest common divisor $d$ of a set $S \subset R$, not all $x \in S$ zero, is a common divisor of the elements of $S$ with the property that any common divisor of all $x \in S$ divides $d$. If for any $S \subset R$ with not all $x \in S$ zero such a $d$ exists, $R$ is called a principal ideal domain (cf. Principal ideal ring). Examples of such domains are the ring $\mathbb{Z}$ of rational integers or polynomial rings $F[X]$, where $F$ is a field (e.g., $\mathbb{C}$ or $\mathbb{R}$ or $\mathbb{Q}$). It is known that a principal ideal domain is also a unique factorization domain. If a greatest common divisor exists for all finite sets $S$, then $R$ is a Bezout domain.

References

[a1]	B.L. van der Waerden, "Algebra" , 1 , Springer (1967) (Translated from German)