The linear congruence equations are the ancient and significant research contents. Most discussions of the linear congruence equations focus on the special cases, for example, there is a linear congruent theorem for solving the congruent linear equation in one unknown and the Chinese remainder theorem for solving the simultaneous congruent linear equations in one unknown, which are involved to find a special solution using the properties of the integer number, and some papers discuss the equation in unknowns. But all the results are not convenient and efficient to solve the equations and not adapt to solving the general system of congruent linear equations in n unknowns. There is no uniform, convenient and efficient technique and theory for the general system of congruent linear equations, like the theory of linear equations over real numbers. The inspiration arose from the elimination of variables when solving the linear equations over real numbers, Generalized the elementary transformations of matrix over real numbers to the integer numbers modulo m, the paper discussed the properties of the modular matrix under the elementary transformations and a similar equivalent transforming theorem for matrix modulom theorem was obtained that any matrix modulo m can be transformed into a canonical diagonal form by means of a finite number of elementary row and column operations. furthermore, by means of the equivalent transforming theorem, the solution criterion theorem and structure theorem were proposed for the general congruent linear equations based on the modular matrix transformations, which extended the theories of the congruent linear equations, and finally, the detailed steps of the uniform method were given for solving the general system of congruent linear equations based on the elementary transformations of matrix modulo m, it can be easily written out the solutions of the system immediately as determining solutions of the system conveniently by elementary matrix transformations. Analysis and discussions indicate that the results are most valuable in science and the proposed technique for solving the system is convenient, efficient and adaptable.

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In the early 1980s of twenty century, Professor Zhang Zhongfu, an expert in graph theory, raised a question in his research: how many points of intersection of diagonal lines are there inside a regular N polygon, hoping to find out a formula of count. It has been more than 20 years since the question was raised, which has aroused the interest of quite a lot of experts, scholars and those who love mathematics, but still remains unsolved. When N is an odd number, proposition 1 can be derived from formula of counting based on proposition 2, which can also be verified by the programme the author has made. Proposition 1: when N is an odd number, there is no concurrence of 3 or more than 3 diagonal lines of regular N polygon. Proposition 2: when N is an odd number, the number of intersection points of inner diagonal lines of regular N polygon is: &a_n=c^{4}_{n}=frac{1}{24}(n-1)(n-2)(n-3)$ But when N is an even number, it becomes quite complicated. When N are some special even numbers referred to by Chang Jianming, there are some laws: the research is made from the perspective of prime factorization of figures. The writer primarily studies the issue by making use of geometrical drawing and imitation by computer programme, so that the programme is improved and the imitating solutions of the numbers of intersection points of inner diagonal lines of regular N polygon with specific programmes can be solved. The solution is popularizable. The writer has worked out the numbers of intersection points of inner diagonal lines of regular 2N polygon when N 50, if conditions of computers in schools allow. By comparing the counting between drawing and computer imitation, the writer has also found out some interesting laws about the numbers of intersection points of inner diagonal lines of regular N polygon, and has put forward some hypotheses.

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