1×2×3×...×2001+2002×2003×...×4002能被4003整除

Question

1×2×3×...×2001+2002×2003×...×4002能被4003 整除

Answer 1

　　1×2×3×...×2001+2002×2003×...×4002
=1×2×3×...×2001+(4003-2001)×(4003-2000)×...×(4003-1)
=1×2×3×...×2001+4003n+(-2001)×(-2000)×...×(-1)
=1×2×3×...×2001+4003n-2001×2000×...×1
=4003n能被4003 整除
说明：(1)(4003-2001)×(4003-2000)×...×(4003-1)这样2001个括号相乘时,展开除(-2001)×(-2000)×...×(-1)无4003外,其余都有4003与别的括号的数相乘,故写为4003n
(2)+(-2001)×(-2000)×...×(-1)共有2001个负数相乘,为负数.