一、引言整数有哪些分类呢?从倍数情况来看就有质数和合数之分;从倍2的情况来看就有奇数和偶数之分;从倒数的情况来看就有尽除除数、纯循环除数和带纯循环除数之分。题目所指的就是后面这三种除数的不同特征。我对纯循环除数和带纯循环除数研究多年,现有本文成果。研究问题:(1)循环节和被除环的关系;(2)纯循环除数和带纯循环除数的特征;(3)键盘里的奥妙;(4)双向被整除环的求法及其表达式。 I. INTRODUCTION What are the classification of integer? From the case of multiples there is a prime and the sum of points; from the situation of times 2 there are odd and even points; from the reciprocal of the situation there are all to do with the number of divisions, pure Rounding divisor and division with pure rounding divisor. The title refers to the different characteristics of the latter three divisors. My research on pure cycle divisor and pure cycle divisor has been for many years. Research Questions: (1) the relationship between the loop section and the removed ring; (2) the pure cycle divisor and the pure cycle divisor feature; (3) the mystery in the keyboard; (4) the bidirectional divisible loop and its expression .