摘要数学竞赛对开发学生智力、促进教学改革、提高教学水平都有着积极的作用。而数论问题由于其特殊的“整数性”经常出现在各类数学竞赛中，和代数、几何、组合成为竞赛的四大组成部分。数论是关于数的学问，重点研究正整数。现阶段小学的数学内容主要以自然数为主，和数论研究的整数性相符。小学“希望杯”全国数学邀请赛自2003年开展以来，深受学生和学校欢迎。为此，笔者决定研究小学“希望杯”全国邀请赛试题中有关数论方面的问题，通过分析历届竞赛中的数论问题，总结数学竞赛中数论问题的趋势和注意事项，并以此作为今后教学中利用此趋势来培养学生创造性思维能力的依据。46768

Mathematics Competition have a positive effect for the development of students' intelligence, promoting the teaching reform, improving teaching standards. And number theory problems due to its special "integer" of frequently appear in all kinds of math competitions. Algebra, geometry, combination and it combined into an integral part of the four contests. Number theory is learning about numbers, focusing on positive integer. Elementary mathematics at this stage mainly in natural numbers-based, and number theory integer of consistent research. Elementary School "Hope Cup" National Mathematics Invitational since 2003 to carry out, students and schools are welcome to it. For this reason, I decided to study primary school "Hope Cup" National Invitational questions in issues related to number theory by analyzing the previous race in number theory problems, sum mathematics competition in number theory problem trends and considerations, and as a future teaching use this tendency to develop on the basis of students' creative thinking ability.

毕业论文关键词：数学竞赛； 数论； 整除理论；同余理论

Keyword: Mathematics competition； Number theory； Divisible theory；Congruence theory

目    录

1.前言 1

2.研究背景 1

3.竞赛中常见的数论问题 2

3.1整除理论有关 2

3.1.1被某个数整除问题 2

3.1.2因数倍数有关问题 3

3.1.3质数有关 3

3.1.4奇偶分析法 4

3.1.5抽屉原理 5

3.2余数理论 5

3.2.1星期几问题 5

3.2.2周期问题 5

3.2.3个位数字问题 6

3.2.4孙子定理相关 6

3.2.5同余和剩余类问题 7

3.3不定方程 7

4.数论问题总结分析 8

4.1总体情况 8

4.2关于整除理论 9

4.3关于余数理论 9

4.4关于不定方程 9

4.5今后趋势 9

5.在教学活动中的建议 10

1.前言 

数学竞赛对开发学生智力、促进教学改革、提高教学水平都有着积极的作用。这项活动激励着学生学习数学的兴趣，培养和提高他们的创造性思维能力［1］。其教育功能显示出这项活动已成为数学教育的重要组成部分。