n is integer, r is the remainder when (n-1)(n+1) is divided by 24. r=?
24 = 2*2*2*3
so (n-1)(n+1) must have 2,2,2,3 i.e 8 & 3 as its factors to be evenly divisible by 24 or else it will give a remainder.
1) 2 is not a factor of n
i.e n is odd so both n-1 & n+1 will be even & since the two nos are consecutive multiples of 2, so if one is divisible by 2 one time, the other will be divisible by 2, 2 times
hence (n-1)(n+1) will have a factor of 8
still we cant say that 3 is one of the fators as n itself may be an odd multiple of 3
2) 3 is not a factor of n
i.e n gives a reamainder of 1,2 when divided by 3
in that case either n-1 or n+1 will have 3 as factor
but we still cant say that this product is divisible by 8
Combine
(n-1)(n+1) is divisible by 8
&
(n-1)(n+1) is divisible by 3
so (n-1)(n+1) is divisible by 24
r =0 SUFF
C
Regards
Samir
