p and q are positive integers. Is p^2 - q^2 divisible by 24?

Question

p and q are positive integers. Is  p^2 - q^2  divisible by 24?

(1) p>q>3
(2) p and q are prime numbers

TestPrepUnlimited · Answer

Is  p^2 - q^2 = (p - q)(p + q)  divisible by 24?

Statement 1:

Insufficient.

Statement 2:

To start off, there are many "no" cases by allowing q = 2 so we focus on finding a "yes" case.

To be divisible by 24 we need  p^2 - q^2 = (p - q)(p + q)  to have a factor of 3 and a factor of 8, for most cases p and q are both odd, which gives the superficial conclusion that  (p - q)(p + q)  is even*even which has a factor of 4. We can note that  2q  is even but not a multiple of 4, so we can express 2q as  4*i + 2 . Then if  p - q  is even but not a multiple of 4, then we add 2q to get  p + q  which must be a multiple of 4. If  p - q  is a multiple of 4, then  p + q  is even but not a multiple of 4. Hence either way we will be able to extract a multiple of 8 from  (p - q)(p + q) .

Now we need either p - q or p + q to be a multiple of 3 now to be divisible by 24. We can try p = 17 and q = 11 for example and see  (17-11)(17+11) = 6*28  has a factor of 24, so insufficient since we have both "yes" and "no" cases.

Combined:

We concluded that  p^2 - q^2  must have a multiple of 8, we want to check if it must have a multiple of 3 now given the conditions. Start off with p, p is expressed as either  3i + 1  or  3i + 2 . q is also expressed as either  3j + 1  or  3j + 2 . Thus when we add/subtract any of the two, we will always achieve  3i - 3j  from p - q, or  3i + 3j + 3  from p + q. Then  p^2 - q^2  will always contain a factor of 3, given statement 1 is true. Thus we can conclude  p^2 - q^2  must have a factor of 24.

Ans: C

MalachiKeti · Answer

Bunuel   KarishmaB  @scottargettestprep pls help with a solution

Bunuel · Answer

p and q are positive integers. Is  p^2 - q^2  divisible by 24?
 
(1) p > q > 3

If p = 5 and q = 4, then p^2 - q^2 = 25 - 16 = 9, which is not divisible by 24.
If p = 4 and q = 4, then p^2 - q^2 = 0, which is divisible by 24.

Not sufficient.

(2) p and q are prime numbers

If p = 3 and q = 2, then p^2 - q^2 = 9 - 4 = 5, which is not divisible by 24.
If p = 7 and q = 5, then p^2 - q^2 = 49 - 25 = 24, which is divisible by 24.

Not sufficient.

(1)+(2) Every prime number greater than 3 can be expressed as either 6k + 1 or 6k - 1, for some integer k.

If both p and q are of the form 6k + 1, then p^2 - q^2 becomes 12(m - n)(3m + 3n + 1). In this case, whether m and n are both even or both odd, the expression is divisible by 24.

If both p and q are of the form 6k + 1, then p^2 - q^2 = (p - q)(p + q) = (6m + 1 - 6n - 1)(6m + 1 + 6n + 1) = 6(m - n)2(3m + 3n + 1) = 12(m - n)(3m + 3n + 1). If m and n are both even or both odd, then m - n is even, and if one is even and the other is odd, then 3m + 3n + 1 is even. Thus, 12(m - n)(3m + 3n + 1) = 12 * 2 * something = 24 * something, so divisible by 24.

If one of p and q is of the form 6k + 1 and the other is 6k - 1, then p^2 - q^2 = (p - q)(p + q) = (6m + 1 - 6n + 1)(6m + 1 + 6n - 1) = 2(3m - 3n + 1)6(m + n) = 12(3m - 3n + 1)(m + n). If m and n are both even or both odd, then m + n is even, and if one is even and the other is odd, then 3m - 3n + 1 is even. Thus, 12(3m - 3n + 1)(m + n) = 12 * 2 * something = 24 * something, so divisible by 24.

Sufficient.

Answer: C.

Hope it's clear.

P.S. Pure algebraic questions are no longer a part of the  DS syllabus  of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

Word Problems 
 Work Problems 
 Distance Problems 
 Mixture Problems 
 Percent and Interest Problems 
 Overlapping Sets Problems 
 Statistics Problems 
 Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check  GMAT Syllabus for Focus Edition

You can also visit the  Data Sufficiency forum  and filter questions by  OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources  to see the types of questions currently tested on the GMAT.

So, you can ignore this question.

Hope it helps.

p and q are positive integers. Is p^2 - q^2 divisible by 24?

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GMAT Data Sufficiency (DS) Questions

p and q are positive integers. Is p^2 - q^2 divisible by 24?

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