binaras 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

Statement 1: n is not divisible by 2
Options for n:
1, 3, 5, 7, 9, 11, 13...

If n=1, then dividing (n+1)(n-1) by 24 yields the following:
(1+1)(1-1) / 24 = 0/24 = 0 R0.
If n=3, then dividing (n+1)(n-1) by 24 yields the following:
(3+1)(3-1) / 24 = 8/24 = 0 R8.

Since R can be different values, INSUFFICIENT.

Statement 2: n is not divisible by 3
Options for n:
1, 2, 4, 5, 7, 8...

If n=1, then dividing (n+1)(n-1) by 24 yields the following:
(1+1)(1-1) / 24 = 0/24 = 0 R0.
If n=2, then dividing (n+1)(n-1) by 24 yields the following:
(2+1)(2-1) / 24 = 3/24 = 0 R3.

Since R can be different values, INSUFFICIENT.

Statements combined:
Options for n:
1, 5, 7, 11...

If n=1, then dividing (n+1)(n-1) by 24 yields the following:
(1+1)(1-1) / 24 = 0/24 = 0 R0.
If n=5, then dividing (n+1)(n-1) by 24 yields the following:
(5+1)(5-1) / 24 = 24/24 = 1 R0.
If n=7, then dividing (n+1)(n-1) by 24 yields the following:
(7+1)(7-1) / 24 = 48/24 = 2 R0.
If n=11, then dividing (n+1)(n-1) by 24 yields the following:
(11+1)(11-1) / 24 = 120/24 = 5 R0.

In every case, R=0.
SUFFICIENT.

The correct answer is C.

Alternate approach:

Statement 1: 2 is not a factor of n.
Thus, n = odd.
Thus, (n-1)(n+1) = the product of two consecutive even integers.
Of every two consecutive even integers, exactly one is a multiple of 4.
Thus, the product of 2 consecutive even integers = the product of an even integer and a multiple of 4 = a multiple of 8.
Since a multiple of 8 can be a multiple of 24 (in which case r=0) or not be a multiple of 24 (in which case r≠0), INSUFFICIENT.

Statement 2: 3 is not a factor of n
Since one of every 3 consecutive integers is a multiple of 3, and n is not a multiple of 3, either (n-1) or (n+1) must be a multiple of 3.
Thus, (n-1)(n+1) = a multiple of 3.
If (n-1)(n+1) is also a multiple of 8, then (n-1)(n+1) = a multiple of 24, in which case r=0.
If (n-1)(n+1) is not a multiple of 8, then (n-1)(n+1) ≠ a multiple of 24, in which case r≠0.
INSUFFICIENT.

Statements 1 and 2 combined:
Since (n-1)(n+1) = a multiple of 8, and either n-1 or n+1 must be a multiple of 3, (n-1)(n+1) = a multiple of 24.
When a multiple of 24 is divided by 24, r=0.
SUFFICIENT.

The correct answer is C.

Thanks

binaras wrote:Hi,

Need some assistance with the following question.

Question
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

Thanks

Solution:

This is a great question for choosing convenient numbers for the variables. When setting up this question, we can also use the difference of squares since we know that (n-1)(n+1) is equal to n^2 - 1. Setting up the question into a mathematical expression, we can say:

(n^2 - 1)/24, Remainder = ?

Statement One Alone:

n is not divisible by 2

This tells us that n is an odd number. Thus, we can try a few different odd numbers, starting with 1. When n = 1, n^2 - 1 = 0. The remainder of 0/24 is 0.

Now let's use n = 3.

When n = 3, n^2 - 1 = 8. The remainder of 8/24 is 8.

Because we have found two different remainders (or values for r), statement one alone is insufficient. We can eliminate answers A and D.

Statement Two Alone:

n is not divisible by 3

This tells us that n cannot be a multiple of 3. Since we have already tested "1," a non-multiple of 3, in statement one, we can again use that value in statement two.

When n = 1, n^2 - 1 = 0. The remainder of 0/24 is 0.

Now let's use n = 2.

When n = 2, n^2 - 1 = 3. The remainder of 3/24 is 3.

Again we again see that we have found two different remainders (or values for r); statement two alone is insufficient. We can eliminate answer B.

Statements One and Two Together:

We know that n cannot be even, nor can it be a multiple of 3. We know that 1 is a possible value for n and that when n = 1, 1^2 - 1 = 0. The remainder of 0/24 is 0.

Next we can use n = 5 (we can't use 2, 3 or 4 since they are either even or a multiple of 3).

When n = 5, n^2 - 1 = 24. The remainder of 24/24 is 0.

Now it appears we have found a pattern that, when we continue to plug in values, fulfills statements one and two: we will get a remainder of zero; however, let's test one more value of n for good measure.

Let's use n = 7.

When n = 7, n^2 - 1 = 48. The remainder of 48/24 is 0.

The answer is C