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If n is a positive integer, is n^2 - 1 divisible by 24?

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Re: If n is a positive integer, is n^2 - 1 divisible by 24?  [#permalink]

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New post 12 Oct 2017, 21:16
I found my answer in the thread. Got it! Thanks again.
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Re: If n is a positive integer, is n^2 - 1 divisible by 24?  [#permalink]

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New post 12 Oct 2017, 21:17
sepehr23 wrote:
Great :)! Thanks for the quick answer.
Consequently, the final result is that "The combined answer is sufficient to say that 'n^2 - 1' is NOT divisible by 24, right?"


Please read carefully: "We have that (n-1)(n+1) is divisible by both 3 and 8 so (n-1)(n+1) IS divisible by 3*8=24. Sufficient."
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Re: If n is a positive integer, is n^2 - 1 divisible by 24?  [#permalink]

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New post 27 Nov 2017, 19:16
kunalbh19 wrote:
If n is a positive integer, is n^2 - 1 divisible by 24?

(1) n is a prime number.
(2) n is greater than 191


We need to determine whether (n^2 - 1)/24 = integer. Notice that 24 is 2^3 x 3, or 8 x 3.

Since n^2 - 1 = (n + 1)(n - 1), when n is odd and not a multiple of 3, we will have the product of two consecutive even integers, one of which is a multiple of 3, and thus n^2 - 1 is divisible by 24. For instance, when n is 5, we have 4 x 6, which is divisible by 24.

Statement One Alone:

n is a prime number.

When n is 5, we see that 24/24 = integer; however, when n = 2, 3/24 does not equal an integer. Statement one is not sufficient to answer the question.

Statement Two Alone:

n is greater than 191.

If n = 192, then n^2 - 1 will be odd and will not be divisible by 24. If n = 199, then n^2 - 1 = (199 + 1)(199 - 1) = 200 x 198 is divisible by 24, since 200 is divisible by 8 and 198 is divisible by 3. Statement two is not sufficient to answer the question.

Statements One and Two together:

From both statements, we see that n is a prime that is greater than 191, and thus it satisfies the case that n is odd and not a multiple of 3. So, n^2 - 1 will be divisible by 24.

Answer: C
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Re: If n is a positive integer, is n^2 - 1 divisible by 24?  [#permalink]

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New post 27 May 2018, 23:20
The question is asking , is n^2-1 divisible by 24

So we can write :
n^2 -1 = 24 I (Where I is an Integer)
OR n^2 = 24I + 1

24I will always be even as 24 is even and irrespective of I being ODD or EVEN , this term will always be EVEN
1 is an odd number .
hence 24I+1 essentially means EVEN + ODD and thus in turn means ODD

So n^2 = ODD , therefore n must be ODD
So the question boils down to finding out is N ODD

A. n is a prime number .
So n can be 2 or 3 .
Hence we cant conclusively say that its ODD , hence "A" is ruled out

B. n >191
again we cant conclusively say that its ODD , hence "B" is ruled out

Now if we combine , we get n is a prime number and greater than 191 .
All Prime numbers greater than 2 are always ODD . Therefore we can conclusively say that N is odd

Hence answer is Option "C"
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Re: If n is a positive integer, is n^2 - 1 divisible by 24? &nbs [#permalink] 27 May 2018, 23:20

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