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

Question

(1) x is odd
(2) x leaves a remainder when divided by 3

LevanKhukhunashvili · Answer

Dont' agree with OA

We are asked if (x-1)(x+1) is divisible by 3* 2^3 
2 stm --> x=2 leaves a reminder when divided by 3 and (2-1)(2+1) is not divisible by 24; x=5 (5-1)(5+1) is divisible by 24

Imo answer is C

nick1816 · Answer

Wrong OA
when x=4, x^2-1=15( not divisible by 24)
when x=7, x^2-1=48( divisible by 24)
Hence B is insufficient 
Answer must be C

chetan2u · Answer

If x is a positive integer,is  x^2 -1 divisible by 24?
1.x is odd
 If x is odd, then x-1 and x+1 will be divisible by atleast a 2 and a 4. so  x^2-1 , which is (x-1)(x+1) will be divisible by 2*4 or 8 at least.
Now, if any one of x-1 or x+1 is divisible by 3, answer is yes, x^2-1 is divisible by 8*3. But, if x is divisible by 3, answer is no.

2.x leaves a remainder when divided by 3
 Now  x^2-1=(x-1)(x+1) , so if x is not divisible by 3, either x-1 or x+1 will be divisible by 3. Therefore, if x is even, the answer will be no, as both x-1 and x+1 will be odd. for example, 2, 4, or 8. Where as if x is odd, for example 7, 11 etc, answer is yes

Combined..
Statement I tells us that  x^2 -1 divisible by at least 8, and Statement II tells us that  x^2 -1 divisible by at least a 3, so  x^2 -1 is surely divisible by at least 8*3 or 24.
Suff

C

axezcole · Answer

Apologies ,marked Option B by mistake.The correct answer is C.

Archit3110 · Answer

#1
x=1,3,5 we get variation in answers, insufficient
#2
x=4,5,7
again in sufficient
from1 & 2 
x=5 issufficient
IMO C

exc4libur · Answer

x≥1 (positive integer)
x^2-1=(x+1)(x-1)
factors(24)=2^3*3

(1) insufic
x=1: 0(2)/24=div
x=3: 2(4)/24≠div

(2) insufic
x=1: 0(2)/24=div
x=2: 1(3)/24≠div

(1/2) sufic
x=1: 0(2)/24=div
x=5: 4(6)/24=div
x=7: 6(8)/24=div

Ans (C)

xyz12345678 · Answer

Even if i take both together, won't it fail at x=1?Ideally it both will onlly true together if x>5

Bunuel · Answer

x = 1 is a valid case. It’s odd and leaves a nonzero remainder when divided by 3. So, for this case x^2 - 1 = 1 - 1 = 0, which is divisible by 24. So this case gives a YES, just like other valid x. There's no contradiction.

ZERO:

1. Zero is an  INTEGER .

2. Zero is an  EVEN  integer.

3. Zero is  neither positive nor negative  (the only one of this kind)

4. Zero is  divisible by EVERY integer except 0 itself  ( \frac{0}{x} = 0 , so 0 is a divisible by every number, x).

5. Zero is a  multiple of EVERY integer  ( x*0 = 0 , so 0 is a multiple of any number, x)

6. Zero is  NOT a prime number  (neither is 1 by the way; the smallest prime number is 2).

7. Division by zero is NOT allowed:  anything/0 is undefined .

8. Any non-zero number  to the power of 0 equals 1  ( x^0 = 1 )

9.  0^0  case is  NOT tested on the GMAT .

10. If the exponent n is positive (n > 0),  0^n = 0 .

11. If the exponent n is negative (n < 0),  0^n  is undefined, because  0^{negative}=0^n=\frac{1}{0^{(-n)}} = \frac{1}{0} , which is undefined.  You CANNOT take 0 to the negative power .

12.  0! = 1! = 1 .

Also, 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.

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

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GMAT Data Sufficiency (DS) Questions

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

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