请教一道GWD上的数学题

cooldavy

If n is a positive integer and ris the remainder when (n – 1)(n + 1) is divided by 24, what isthe value of r?
(1)     2 isnot a factor of n.
(2)     3 isnot a factor of n.


答案是C，虽然枚举了一下貌似可以，但是不知道怎么推导的，求详解

wolfheulen

（1），（2）分别不充分，这个比较容易看出来，可以举出反例

（1）（2）在一起，可以推断出(n – 1)(n + 1) 能被24整除，证明如下：
if 2 is not a factor of n, then 2 is a factor of n-1, and 2 is a factor of n+1

furthermore, 4 is a factor of one of n-1, n+1. suppose n-1 is not divisible by 4. since it is divisible by 2, its remainder when divided by 4 must be 2. then n+1's remainder when divided by 4 must be 0. similarly, we can conclude that if n+1 is not divisible by 4, then n-1 must be divisible by 4.

therefore (n – 1)(n + 1) can be divided by 2*4=8.

also, since 3 is not a factor of n, then 3 must be a factor of either n-1 or n+1. (if the remainder of n/3 is 1, then n-1 is divisible by 3. if the remainder of n/3 is 2, then n+1 is divisible by 3)

therefore (n – 1)(n + 1) can also be divided by 3

since (n – 1)(n + 1) can be divided by 3 and 8, it is divisible by 24.

hope that helps

birdflyorc

（1）2不是n的因数，说明n是个奇数，那么(n-1)(n+1)就是个连续偶数，两个连续偶数一个肯定是2的倍数一个肯定是4的倍数，继而能够推出（n-1）（n+1）能够被8整除，但是否能被24整除不清楚
（2）3不是n的因数，（n-1）（n+1）中肯定有一个是3的倍数，但不知道是不是24的倍数
（1）+（2）推出（n-1）（n+1）是24的倍数；那么r就为0了

中文版本

cooldavy

谢谢楼上几位大哥的热心解答~
总算是明白了