在初、高中教材和一些初等数学参考书中,经常遇到一些关于整数的整除性证明的问题。本文就这一问题给出五种证明方法。一般来说,有关整数的整除性的问题,都可以用这些方法来证明。一用分解因式法证明例1 已知m是自然数,求证m~5-5m~4+4m能被120整除。证明:m~5-5m~4=m(m~2-1)(m~2-4) =m(m-2)(m-1)(m+1)(m+2) ∴m~5-5m~4+4m可化成五个连续的整数m-2,m-1,m,m+1,m+2的乘积的形式。从而知,原式能被5整除,又能被3整除。另一 In elementary and high school textbooks and some elementary math reference books, problems with the divisibility of integers are often encountered. This article gives five ways to prove this problem. In general, the problem of integer divisibility can be proved by these methods. One uses the factorization method to prove that example 1 Known that m is a natural number, verify that m~5-5m~4+4m can be divisible by 120. Proof: m~5-5m~4=m(m~2-1)(m~2-4) =m(m-2)(m-1)(m+1)(m+2) ∴m~ 5-5m~4+4m can be converted into a product of five consecutive integers m-2, m-1, m, m+1, m+2. Thus, the original formula can be divisible by 5 and divisible by 3. another