r=n^2-(24k+1)..where k is any integer

if n^2<24..n=1,2,3,4 and r=0,3,8,15
if n^2>24..n=5,6,7,8,9,10,11,12.,13,14.. and r=0,11,0,15,8,3,0,23,0,1

clearly neither of the statements alone nor together are sufficient to get a certain value of r

option E should be the ans

ashforgmat ---

Whatever given in the question just try to write into conditions, here,
Conditions:
1. n is +ve integer
2. (n+1)(n-1) is divisible by 24 and the remainder is r. we can write as,
(n+1)(n-1) = 24*k + r here k = quotient and r = remainder.
I can also write this as,
(n^2 ) - 1 = 24*k + r

Question asked for the value of 'r' - remainder ??? Since it was a value-based question, we should have a UNIQUE ANSWER to tell the statement is sufficient or not. Let's go through the statements one by one.

Statement 1
The given condition here is n should be divisible by 2
Lets take different values for n and check whether r value is UNIQUE or NOT.
If n = 6 which is divisible by 2 and Now check what is the value of r,
n^2 - 1 = 36 - 1 = 35, we write this number as 24*1+11, so here r=11
In such a way take another +ve integer which is divisible by 2,
if n=10, then r = 3,
As r is not having UNIQUE value, Hence, Statement-1 is INSUFFICIENT.

Statement 2
The given condition here is n not should be divisible by 3
If n = 10 which is not divisible by 3, then (n^2)-1= 100-1=99, we can write it as 24*4+3 then r = 3
If n = 11 which is not divisible by 3, then (n^2)-1=121-1=120, we can write it as 24*5+0 then r=0
As r is not having UNIQUE value, Hence, Statement-2 is also INSUFFICIENT.

BOTH
n should not divisible by 3, but should divisible by 2.
If n=8 which is not divisible by 3, but divisible by 2, then (n^2)-1=64-1=63, we can write it as 24*2+15 then r=15
If n=10 which is not divisible by 3, but divisible by 2, then (n^2)-=100-1=99, we can write it as 24*4+3 then r = 3
As r is not having UNIQUE value, Hence, BOTH the statements are also INSUFFICIENT.

Hence, the correct answer is E

HTH, GOOD LUCK,

Thanks,
Rajesh,
Loves GMAT....!!!!