同余問題的討論-應(yīng)用數(shù)學(xué)畢業(yè)論文(已修改)
2025-06-06 14:29 本頁(yè)面
【正文】 南 京 師 范 大 學(xué) 泰 州 學(xué) 院 畢 業(yè) 論 文（設(shè) 計(jì)） （ 一 五 屆） 題 目： 同余問題的討論 院（系、部）： 數(shù)學(xué)科學(xué)與應(yīng)用學(xué)院 專 業(yè)： 數(shù)學(xué)與應(yīng)用數(shù)學(xué) 姓 名： 朱嘉斌 學(xué) 號(hào) 08110244 指導(dǎo)教師： 黃玉才 南京師范大學(xué)泰州學(xué)院教務(wù)處 制 南京師范大學(xué)泰州學(xué)院本科畢業(yè)論文 1 摘要： 無論是從理論上還是從應(yīng)用上來看 , 同余理論都是數(shù)論中重要的一部分 , 初等數(shù)論的核心是以同余的性質(zhì)、 同余的概念為基礎(chǔ)發(fā)展起來的同余理論 , 同余理論是數(shù)論所特有的思想 , 也是研究余數(shù)除法的重要工具。 本文首先歸納總結(jié)了同余的性質(zhì)及其應(yīng)用 , 在同余的定義的基礎(chǔ)上 , 介紹了同余的中幾個(gè)重要的性質(zhì) , 并且側(cè)重討論了同余的幾個(gè)重要的應(yīng)用 , 剩余類和剩余系也是它的研究對(duì)象。然后 ,又歸納總了同余方程的解及應(yīng)用 , 例如 : 一次同余式的形式、解同余式以及解一次同余式組和中國(guó)剩余定理的應(yīng)用等問題 , 討論的過程中由淺入深 , 層層推進(jìn) , 對(duì)相關(guān)知識(shí)的總結(jié)形成了較為完整的知識(shí)體系 , 對(duì)解同余方程的一 些特殊的性質(zhì)理解更為深刻。再次 , 結(jié)合同余理論的概念與性質(zhì) , 給出常見的習(xí)題及解題技巧 , 并舉例說明了幾個(gè)重要的同余定理和它們的應(yīng)用。最后， 余數(shù)問題是數(shù)論知識(shí)板塊中另一個(gè)內(nèi)容豐富，題目難度較大的知識(shí)體系，也是各大杯賽小升初考試必考的奧數(shù)知識(shí)點(diǎn)，所以學(xué)好本講對(duì)于學(xué)生來說非常重要。余數(shù)問題主要包括了帶余除法的定義，三大余數(shù)定理（加法余數(shù)定理，乘法余數(shù)定理，和同余定理），及中國(guó)剩余定理和有關(guān)棄九法原理的應(yīng)用。 關(guān)鍵字：同余；剩余系； 一次同余式 ； 奧數(shù)數(shù)論 的 同余問題 Abstract: both theoretically and from the application point of view,congruence theory is an important part of the number theory, the core of elementary number theory is based on the nature of the congruence, developed on the basis of the concept of congruence congruence theory, the thought of congruence theory is unique to number theory, is also an important tool to study the remainder division. This paper summarized the properties and applications of congruence, on the basis of the definition of congruence, introduced with Yu Dezhong several important properties, and discusses the congruence on several important application of residue class and residue system is its research object. Then, summarized the total congruence equation solution and application, for example: 南京師范大學(xué)泰州學(xué)院本科畢業(yè)論文 2 a congruence type, congruence type and solution in the form of a congruence type group and the application of the Chinese remainder theorem, discuss the process of unit 1, layer upon layer, summary of knowledge formed a relatively plete knowledge system, some special properties of solution congruence equation more profound understanding. Again, the concept of bining the theory of congruence and the nature of mon problems and problem solving skills, and illustrates several important with Chinese remainder theory and their applications. Finally, the remainder is another rich theory knowledge, the subject is difficult knowledge system, is also a major cup among the test study of Olympic math knowledge, so learn the speaking is very important for students. Remainder problem mainly includes the definition of take over division, the three big remainder theorem (remainder theorem of addition, multiplication remainder theorem, and the Chinese remainder theory), and the Chinese remainder theorem and the nine method principle of the application. Key words: congruence。 Residue system。 A congruence type。 Aoshu congruence problems in number theory. 南京師范大學(xué)泰州學(xué)院本科畢業(yè)論文 3 目錄 1 緒論 .................................................... 4 ............................................................................................................... 4 2 同余的性質(zhì)及應(yīng)用 .............................................. 5 同余的概念 ..................................................................................................................... 5 ............................................................................................................... 5 同余式性質(zhì)的應(yīng)用 .................................................................................................... 7 剩余類和完全剩余系 ....................................................................................................... 8 .................................................................................................. 10 ............................................................................................................. 10 .................................................................................................. 10 ......................................................................................................... 12 3 同余方程的解及應(yīng)用 ............
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