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在82年数学通报第10期上,我曾建议中学生学习点初等数论基本知识。在此,我们介绍一些具体有趣的整数问题,这对于熟悉整数问题的思想方法是有益的。我们知道,整数被m除按余数可以分为m类,当m=2时,即分为{2k}(偶数)和{2k+1}(奇数)两大类,而当m=3时,则分为{3k},{3k+1},{3k+2}等三类。中学生对此并不难理解,但对这种分类的应用多数人却是陌生的。本文打算介绍3的剩余类的性质及一些有趣应用。有兴趣的读者不难自己将有关结果推广到m=4,5,…及一般剩余类的情形,再进一步学习整除理论、同余理论及初等数论中一些重要知识,就不会感到抽象费解了。 (一) 整数被3除的余数运算规律我们用余数0、1、2分别表示类{3k}、
In the eighth issue of the Mathematics Bulletin of the 82nd year, I had suggested that middle school students learn the basics of elementary number theory. Here, we introduce some specific interesting integer problems, which is beneficial to the idea of the integer problem. We know that an integer divided by m can be divided into m classes by the remainder. When m=2, it is divided into two categories: {2k} (even) and {2k+1} (odd), and when m=3, It is divided into {3k}, {3k+1}, {3k+2} and other three categories. Middle school students are not hard to understand, but the majority of people who use this classification are unfamiliar. This article intends to introduce the nature of 3 remaining classes and some interesting applications. It is not difficult for interested readers to extend their results to the case of m=4, 5,..., and general residuals, and to further study some important knowledge in the theory of divisibility, congruence, and elementary theory of numbers. . (1) The remainder arithmetic division of integers by 3 We use the remainders 0, 1, and 2 to represent classes {3k}, respectively.