1. taking 3,5,7,9, etc i found sum give remainder 0and sum give 1 so not sufficient
2. taking 7,11,13 etc give o remainder but if i take 4 then it give fraction so not sufficient

so imo E

ern5231 wrote:If n is a positive integer and r is the remainder when (n-1) (n +1) is divided by 24, what is the value of r?
1). N is not divisible by 2
2). N is not divisible by 3

This is a value question .

We know n is a positive integer and thus (n-1),n , (n+1) will be consecutive integers .

Now lets look at factors of 24 = 2 x 2 x 2 x 3

We have a find the reminder when (n-1) (n +1) / 24 (considering (n-1) x (n +1) )

1). N is not divisible by 2 .

That means n is not even . Any 3 consecutive numbers will have a multiple of 3 .

Also if n is not even that is n is odd . Then (n-1) and (n+1) will be even (consecutive integers)

That means n could take values such as 3, 9 , 27 ......

So the product (n-1) x (n+1) will have at least 2 - 2's in its prime factors that will cancel out with the factors of 24 . But the reminder will be different in each case .

So Insufficient .

2). N is not divisible by 3 .

This means n is even . Now since n is eve then (n-1) and (n+1) will be odd .

That means we that there will be no 2's in the product of (n-1)x (n+1).

Ex : if n = 2 then (n-1) x (n+1) = 1 X 3 ----- > no 2 's in the product.

But again we are not sure of the reminder in this case as value of reminder be different for different value of n .

Not Sufficient .

Combine 1 and 2 .

n is not divisible by 2 and n is not divisible by 3 .

That means n is a prime number .

Also we know that all prime numbers except 2 are odd number hence (n-1) and (n+1 ) both will be even , that is we will have at least two 2's in its factors .

And out of the two that is (n+1) and (n-1) one will be a multiple of 3 as 3 consecutive numbers will have a multiple of 3 in them . So we have two 2's and a 3 in the product of (n-1) and (n+1).

And any even number great than 2 will have more than one 2 in its prime factors .

So we have at least three 2's and a 3 in the product if (n-1) and (n+1).

these are the factors of 24 . So when (n-1) x (n+1) / 24 will leave 0 reminder .

Hence IMO C.