Euler演段的有效证法及其在解题中的应用

A Verified Proof of "Euler Deduction" and its Application in Solution

本文针对在校大学生在学习《初等数论》《竞赛数学》等课程中产生的困惑的基础上,首先介绍一个奇妙的解决困惑的方法——"Euler演段",同时在给出Euler演段概念的基础上,利用辗转相除法结合数学归纳法对"Euler演段"的正确性给予证明;最后给出了该方法在学习《初等数论》《竞赛数学》中如何求最大公约数、最小公倍数、解k元一次不定方程、解一次同余方程等方面的实际应用.

An Euler deduction is introduced as a miracle method to solve the dilemma encountered by the college students in their study of such courses as ＂ Elementary Number Theory＂ and ＂ Competition Mathematics＂. After an introduction to the concept of Euler deduction,this paper proves the validity of Euler deduction,employing division algorithm in combination with mathematical induction. It is also applied in the students ＇ study of above courses to seek greatest common divisor and lowest common multiple,and to solve K ray indefinite equation and congruence equation.