同余在数学竞赛中的应用

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1、毕　业　论　文题目：同余在数学竞赛中的应用18目录标题······································································1摘要······································································1引言·····································································2181同余的概念······························

2、·······························22同余的基本性质·························································33同余性质在算术里的应用·················································43.1检查因数的一些方法··················································43.2弃九法（验证明整数计算结果的方法）····································53

3、.3同余性质的其他应用··················································63.3.1利用同余理论求余数···············································63.3.2利用同余可以证明整除问题·········································74利用同余性质求简单同余式的解···········································84.1一次同余式、一次同余式解的概念·····················

4、·················8184.2孙子定理解一次同余式组··············································84.3简单高次同余式组及，为质数，的解数及解法的初步讨论··········································94.4简单二次同余式，，解的判断··················11结论·····································································13参考文献················

5、·················································14致谢·····································································15181818181818同余在数学竞赛中的应用【摘要】18同余理论是初等数论的重要组成部分，是研究整数问题的重要工具之一，在处理某些整除性、对整数的分类、解不定方程等方面的问题中有着不可替代的功能。与之相关的数论定理有欧拉定理、费尔马定理、和中国剩余定理。同余是数学竞赛的重要组成部分。本文从数学竞赛这个大范围入手

6、，局限于数论在数学竞赛中的地位和作用，选择同余性质作为切入点，介绍了同余理论的系统知识及同余性质的一些简单应用，并对数学竞赛中有关同余理论的应用作了系统的划分，每一部分都有相关及紧密联系的例题加以举例说明。【关键词】同余同余理论数学竞赛引言数论的一些基础内容的学习，一方面可以加深对数的性质的了解，更深入的理解某些其他邻近学科，另一方面，可以加强数学训练。而整数论知识是学习数论的基础，其中同余理论也是整数论的重要组成部分，所以学好同余理论是非常重要的。18在日常生活中，我们所要注意的常常不是某些整数，而是这些数用某一固定的数去除所得的余数，例如我们问现在是几

7、点钟，就是用24去除某一个总的时数所得的余数；问现在是星期几，就是问用7去除某一个总的天数所得的余数，假如某月2号是星期一，用7去除这月的号数，余数是2的都是星期一。我国古代孙子算经里已经提出了同余式,,…,这种形式的问题，并且很好地解决了它。宋代大数学家秦九韶在他的《数学九章》中提出了同余式,,是个两两互质的正整数，,的一般解法。同余性质在数论中是基础，许多领域中一些著名的问题及难题都是利用同余理论及一些深刻的数学概念，方法，技巧求解。例如，数论不定方程中的费尔马问题，拉格朗日定理的证明堆垒数论中的华林问题，解析数论中，特征函数基本性质的推导等等。在近现

8、代数论研究中，有关质数分布问题，如除数问题，圆内格点问题，等差级数