If n is a positive integer and r is the remainder when n^2 - 1 is

I like the approach in the post above best for this problem; when you see something like n^2 - 1 in a GMAT question, it will almost always be useful to use the difference of squares factorization: n^2 - 1 = (n+1)(n-1). A less elegant alternative is to write n = 2k + 1. Then n^2 - 1 = (2k + 1)^2 - 1 = 4k^2 + 4k + 1 - 1 = 4k^2 + 4k = 4(k)(k + 1), and since k and k+1 are consecutive integers, one of them must be divisible by 2, so 4(k)(k + 1) must be divisible by 4*2 = 8.

To answer the question in the original post, if n=1, then n^2 - 1 = 0. So the question becomes, what is the remainder when 0 is divided by 8? Well, 0 is divisible by every positive integer; the quotient is zero and the remainder is zero. If you think back to how you first learned division, this should hopefully be clear: if you have, say, 11 apples and 8 children, we can give each child 1 apple and we have 3 left over, so the quotient is 1 and the remainder is 3 when you divide eleven by eight. If we have 0 apples and 8 children, we can give each child 0 apples and we have 0 left over, so the quotient is 0 and the remainder is 0 when we divide zero by eight.

Thanks for the great help.

But I still feel very confused.
The question is asking "what's the value of r"?
I understand when n is 3,5,7, the r will be 1. However, if n is 1, r will be 0. In this case, we have two answers for r and we can't really tell the exact value for r, right? This is the reason why I don't think the first one work and the answer should be "E". Am I in the right path? Thanks for the help again.

No, the remainder will be zero for any of the values 1, 3, 5, or 7 (or for any other odd value of n). If, say, n=5, then n^2 - 1 = 25 - 1 = 24, and the remainder when we divide 24 by 8 is zero.

If n is a positive integer and r is the remainder when n^2 - 1 is divided by 8, what is the value of r?

(1) n is odd
(2) n is not divisible by 8



As the OA is not provided this is how I solved this. Please let me know whether I am right or not.

Considering Statement 1

n =1 then remainder will be zero. ----------------> Is my thinking correct over here? If it is then for me this statement is sufficient to answer the question.

Considering statement 2

There will be different values for r and therefore its insufficient.

Therefore, for me the answer should be A unless someone has any other ideas. Please help.

As you can see from my post above answer is indeed A. But you arrived to A while trying only one value for n, which is not enough. For some other question, different values could give different answers and a statement would be insufficient in that case. What I mean is, when you decide that a statement is sufficient based only on plug-in method you should make sure that you tried several different numbers (and saw some pattern maybe), and even in this case you may not be 100% sure that the answer would be correct. Though if several numbers give the same answer and you are able to see some pattern, then you can make an educated guess that a statement is sufficient and move-on.

Generally on DS questions when plugging numbers, your goal is to prove that the statement is not sufficient. So you should try to get an YES answer with one chosen number(s) and a NO with another.

Hope it's clear.

Bunuel - its clear apart from one minor doubt. In statement 1 why can't we take n =1 ?

Actually we can. Solution above does not exclude this possibility: n=1=odd then n^2-1=0 and zero is divisible by any integer (except zero itself), so it's divisible by 8 too. Sufficient.

The point is that you arrived that (1) is sufficient based only on one value of n, n=1. And as I discussed above one value is not enough to conclude that the statement is sufficient.

Hope it helps.

Please check the solution in the file attached

Given: r is the remainder when (n² - 1) is divided by 8

Target question: What is the value of r?

Statement 1: n is odd
Let's test some ODD values of n
If n = 1, then n² - 1 = 1² - 1 = 0, and 0 divided by 8 leaves remainder 0. So, the answer to the target question is r = 0
If n = 3, then n² - 1 = 3² - 1 = 8, and 8 divided by 8 leaves remainder 0. So, the answer to the target question is r = 0
If n = 5, then n² - 1 = 5² - 1 = 24, and 24 divided by 8 leaves remainder 0. So, the answer to the target question is r = 0
If n = 7, then n² - 1 = 7² - 1 = 48, and 0 divided by 8 leaves remainder 0. So, the answer to the target question is r = 0
At this point, we might conclude that r will ALWAYS be 0
So, statement 1 is SUFFICIENT

----ASIDE--------------------------------
If you're not convinced, here's an algebraic solution as well:
If n is ODD, then n = 2k + 1 (for some integer value of k)
So, n² - 1 = (2k + 1)² - 1 = 4k² + 4k + 1 - 1 = 4k² + 4k = 4(k² + k)

Notice that, if k is odd, then k² + k is EVEN, which means k² + k = 2 times some integer
So, n² - 1 = 4(k² + k) = 4(2 times some integer) = 8 times some integer
In other words, n² - 1 is a multiple of 8, which means the answer to the target question is r = 0

Similarly, if k is even, then k² + k is EVEN, which means k² + k = 2 times some integer
So, n² - 1 = 4(k² + k) = 4(2 times some integer) = 8 times some integer
In other words, n² - 1 is a multiple of 8, which means the answer to the target question is r = 0

In both cases, the answer to the target question is r = 0
Since we can answer the target question with certainty, statement 1 is SUFFICIENT
------------------------------------------

Statement 2: n is not divisible by 8
There are several values of n that satisfy statement 2. Here are two:
Case a: n = 3. In this case, n² - 1 = 3² - 1 = 8, and 8 divided by 8 leaves remainder 0. So, the answer to the target question is r = 0
Case b: n = 4. In this case, n² - 1 = 4² - 1 = 15, and 15 divided by 8 leaves remainder 7. So, the answer to the target question is r = 7
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

Asked: If n is a positive integer and r is the remainder when n^2 - 1 is divided by 8, what is the value of r?

(1) n is odd
n = 2m + 1
n^2 - 1 = (2m+1)^2 - 1 = 4m^2 + 4m = 4m(m+1)
The remainder when n^2 - 1 is divided by 8 = 0 = r
SUFFICIENT

(2) n is not divisible by 8
If n= 5; n^2 - 1 = 24; r = 0
If n=6; n^2 - 1 = 35; r = 3
NOT SUFFICIENT

IMO A

Hi,
What if n = 1, n-1 = 0, n+1 = 2? So how will (n-1)(n+1) be divisible by 8?

Yes.

Zero is divisible by every integer (except zero itself, as division by zero is not allowed).

Integer \(a\) is divisible by integer \(b\) means that \(\frac{a}{b}=integer\) (so remainder is zero).

So, as 0/(any non-zero integer) = 0 = integer, then 0 is divisible by any non-zero integer, including 8.

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