If p is a positive integer, what is the remainder when p2-1 is divide

Question

If  p  is a positive integer, what is the remainder when  p^2-1  is divided by 24?

(1)  p  is a prime number greater than 19
(2)  p  is an odd integer that is not a multiple of 3

IanStewart · Accepted Answer

p-1, p, and p+1 are three consecutive integers. Since multiples of 3 are three apart, among any three consecutive integers, we must have exactly one multiple of 3. If, as each Statement tells us, p is not a multiple of 3, then one of p-1 or p+1 will be.

If p is odd, as each Statement tells us, then p-1 and p+1 are consecutive even integers. Every second even integer is a multiple of 4, so exactly one of p-1 or p+1 is a multiple of 4 (at least), and the other is a multiple of 2.

p^2 - 1 = (p + 1)(p - 1), and since we can be sure we have a multiple of 3, a multiple of 4, and another multiple of 2 in the product (p+1)(p - 1), this product must be divisible by (3)(4)(2) = 24, and we'll get a remainder of zero when we divide it by 24. So the answer is D.

There's an older GMATPrep question that is very similar to this one.

lucasd14 · Answer

Hi Ian, if p=1 you got p-1 = 0 and p+1 = 2.. in that case the 0 would be the multiple of 4?

IanStewart · Answer

Yes, exactly, and in that case, zero is also the multiple of 3. But in real GMAT divisibility questions, you honestly don't need to spend any time worrying about zero; zero won't produce any exceptions that would ever make you change your answer, and zero is usually disallowed as a possible number anyway (because you're generally told you're dealing with positive integers in divisibility problems).

In other topics (like Algebra, say) you might sometimes need to worry about zero separately, though it depends on how you're solving questions, and a lot of prep company questions seem to be designed to test 'traps' about zero that aren't tested on the actual GMAT, or at least are not in the roughly ten thousand official questions I've seen.

nd7012 · Answer

1. All primes are 6K+/- 1 format. Square will give -1^2/24 =1 and 1/24=1. Thus reminder 0.
2. Not a Multiple of 3 is 1,5,7,11,13 . Thus all will be prime. Again same format 6k+/- 1. Thus will give reminder 1. Final is 1-1 =0. 
Thus D

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sevenfive · Answer

Small correction with Statement 2 - All will not be prime 
1 is not a prime 
25 is not prime

bumpbot · Answer

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If p is a positive integer, what is the remainder when p2-1 is divide

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If p is a positive integer, what is the remainder when p2-1 is divide

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