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一个整数a能被质数3整除的特征是:a的各位数字之和能被3整除;能被11整除的特征是:a的奇位数字之和与偶位数字之和的差能被11整除。自然,这只是关于数的整除特征的两个特例。一般地,对于任意质数p(p≠2、5),a能被p整除的特征是什么?我们将看到,a能被p整除的特征与所给特例完全是类似的。即一个数能否被3、11整除的特征并不是孤立的,它反映了关于数的整除特征的两个普遍规律。首先引进循环指标与整数分节的概念。设纯循环小数1÷p的循环节长度(即循环节的位数)为r。为方便计,将r叫做p的循环指标。并将整数a从个位按r位分节。如43256789从个位按三位分节,则有如下三节:
The feature that an integer a is divisible by the prime number 3 is that the sum of the digits of a can be divisible by 3; the feature that can be divisible by 11 is that the difference between the sum of the odd bit numbers of a and the even number can be divisible by 11 . Naturally, this is just two special cases of the divisibility feature of numbers. In general, for any prime number p (p ≠ 2, 5), what are the characteristics of a being divisible by p? We will see that the feature that a can be divisible by p is completely similar to the given special case. That is, the feature of whether a number is divisible by 3, 11 is not isolated, and it reflects two universal laws concerning the divisibility of numbers. First introduce the concept of cyclic indicators and integer segments. Let the length of the loop section (i.e., the number of loop sections) of the pure loop fraction 1÷p be r. For convenience, r is called the cyclic indicator of p. And divide the integer a by r bits from one bit. For example, 43256789 is divided into three sections from the ones place. There are three sections as follows: