jdtomatito wrote:

I think this question has been posted before, but I couldn't find the thread.

n is a positive integer. Find the remainder of (n+1)(n-1)/24

(1) 2 is not a factor of n

(2) 3 is not a factor of n

As other members chose different choices of C and E. I now consider C

(1)(2)

from (2) we know n divided by 3 has remainder of 1 or 2, in any cases, (n-1)(n+1) is divisible by 3.

In order to be divisible by 24, (n+1)(n-1) must be divisible by 8

+from (1) ,we eliminate those n which divided by 8 have remainders of 0,2,4,6,8 because such n is divisible by 2, which conflicts the statement.

+ if n divided by 8 has r (remainder) of 1, then n-1 must be divisible by 8

+ if r=7 ---> n+1 must be divisible by 8

+ if r=3 --> n= 8x+3 ( x is integer) ---> n+1= 8x+4 is divisible by 4

AND n-1 = 8x+2 is divisible by 2. Thus, (n+1)(n-1) is divisible by 8.

+ if r=5 --->n= 8z+5( z is integer) ----> n+1= 8z+6 is divisible by 2

AND
n-1= 8z+4 is divisible by 4 ------> (n+1)(n-1) is divisible by 8

THUS, with (1)and (2) we always have (n+1)(n-1) is divisible by 8...the product is also divisible by 3, SO (n+1)(n-1) is divisible by 8*3= 24

C it is.