规定:=1×2×3×...×n.那么1!×1+2!×2+...90!×90+91!×91除以2002的余数是____.

问题描述:

规定:=1×2×3×...×n.那么1!×1+2!×2+...90!×90+91!×91除以2002的余数是____.

首先,n*n! = (n+1)*n! - n! = (n+1)! - n!
所以,1!*1+2!*2+...+91!*91 =
(2!-1!) + (3!-2!) + ...+(92!-91!) = 91!-1
然后,2002 = 2*7 *11*13
所以91!除以2002的余数是0,减一以后余数是2001