In this paper, the concepts of the least common multiple of polynomial matrices and the prime polynomial matrix are introduced, and some algebraic properties of the greatest common divisor and of the least common multiple of polynomial matrices are given.

The algorithms of basic operations for large integer,addition,subtraction and comparison,are presented based on mixed radix representation,the algorithms of multiplication,division,modulo,and greatest common divisor are conveyed from addition machins.

In this paper,an algorithm,which is called the extended Euclidean algorithm,is derived such that x and y can be simultaneously computed when the greatest common divisor ( a,b ) is computed by the Euclidean algorithm.

A united method for greatest common factor and least common multiple in euclidean ring;

A new method to solve the greatest common factor of several polymerizations on polymerization ring is introduced.

This paper gives a method that through elementary transformming the row of the matrix, the greatest common factor of the Euclidean Ring s factors can be found.

A theorem for any A∈M 2Z (M 2Z is the ring of all integral 2×2 matrices) and any integer m and n , there are X, Y∈M 2Z such that A=X+Y and det X=m , det Y=n if and only if ( m-n ) is divisible by the maximal common factor of all elements of A is proved.

It improved the Greedy-Edge Approximation Algorithm,proved and discussed that the approximation factor of the Improved Greedy-Edge Approximation Algorithm containing the number of Single point Greedy-Edge and the number of Double point Greedy-Edge was not more than 2.

An approximation factor η is suggested to characterize this property.