If p is a positive integer, is p^2 divisible by 96?

Question

(1) p is a multiple of 8.
(2) p^2 is a multiple of 12.

GMATinsight · Accepted Answer

Questions: If p is a positive integer, is p^2 divisible by 96?

96 =  2^5 * 3

i.e. for  P^2  to be a multiple of 96, it must be a multiple of  (2^5 * 3)

Since P is an Integer therefore  P^2  must be a perfect square

i.e.  P^2  MUST be a multiple of (2^6 * 3^2)

i.e P MUST be a multiple of  (2^3 * 3)

Question Stem Modified: Is P a multiple of  (2^3 * 3) ?

Statement 1: p is a multiple of 8

This confirms that p is a multiple of  2^3  but since it's still unknown whether p is a multiple of 3 or not therefore

NOT SUFFICIENT

Statement 2: p^2 is a multiple of 12

This confirms that p is a multiple of 3 and 2 but since it's still unknown whether p is a multiple of  2^3  or not therefore

NOT SUFFICIENT

Combining the two statements

Statement 1 confirms that p is a multiple of  2^3

Statement 2 confirms that p is a multiple of 3

i.e. p is a multiple of  2^3 * 3

SUFFICIENT

Answer: Option  C

EMPOWERgmatRichC · Answer

Hi All,

This question can be solved with a mix of prime-factorization, Number Properties and TESTing VALUES.

We're told that P is a POSITIVE INTEGER. We're asked if P^2 is divisible by 96. This is a YES/NO question.

Before jumping to the two Facts, we can dissect the prompt a bit. 96 = (2^5)(3), so for a number to be divisible by 96, that number MUST have at least (2^5)(3) in its prime-factorization. Since P is an INTEGER and it's P^2 that's divisible by 96, we have to make sure that we're following ALL of the rules in this prompt before we TEST VALUES...

Fact 1: P is a multiple of 8.

8 = (2^3), so P^2 will include (2^6) at the minimum. To be divisible by 96 though, we need to have a '3' in the prime-factorization...

IF....
P = 8, P^2 = (2^6) and the answer to the question is NO

IF....
P = 24, P^2 = (2^6)(3^2) and the answer to the question is YES
Fact 1 is INSUFFICIENT

Fact 2: P^2 is a multiple of 12.

Here, we have an interesting 'wrinkle'....P MUST be an integer, but there is no integer (when squared) that equals 12...The smallest multiple of 12 that is a perfect square is 36...

36 = 6^2 = (2^2)(3^2) --> this has the necessary '3', but not enough 2's...

IF....
P = 6, P^2 = (2^2)(3^2) and the answer to the question is NO

IF....
P = 24, P^2 = (2^6)(3^2) and the answer to the question is YES
Fact 2 is INSUFFICIENT

Combined, we know....
P is a multiple of 8
P is a multiple of 6

From this, we know that P COULD be 24, 48, 72, etc. ALL of those, when squared, will be divisible by 96. The answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer:  C

GMAT assassins aren't born, they're made,
Rich

peachfuzz · Answer

If p is a positive integer, is p^2 divisible by 96?

(1) p is a multiple of 8.
If p^2=96^2 yes
if p^2=16 no
Insufficient

(2) p^2 is a multiple of 12.
if p^2=144 no
if p^2= 144^2 yes
 insufficient

Combined, p^2 has a minimum of five 2s and one 3
sufficient

Answer: C

UJs · Answer

peachfuzz  you need one correction in your solution for stmt #1

UJs · Answer

If p is a positive integer, is p^2 divisible by 96?

Stmt (1)  p is a multiple of 8.
lets pick numbers, take {p=8, p=96}
 
p = 8 ; -------------->  p^2/96^2 = 64/96^2 = No 
p = 96 ; -------------->  p^2/96^2 = 96^2/96^2 = Yes  
  not sufficient

Stmt (2)  p^2 is a multiple of 12.
lets take {p=6, p=96 } ;   Note   we picked p=6 because p is a positive integer, we can't say p^2 = 12; need a perfect square value for p^2;
 
p = 6 ; -------------->  p^2/96^2 = 36/96^2 = No 
p = 96 ; -------------->  p^2/96^2 = 96^2/96^2 = Yes  
  not sufficient

together we know p is a multiple of 8 (p^2 must have 8 * 8 ), and p^2 is a multiple of 12 (so it must have 3 * 3 )
 minimum value p^2 = 8 * 8 * 3 * 3 = 576 
 p^2/96^2 = yes 
  sufficient

Ans :  C

Mo2men · Answer

Hi Rich,
In order to test fact number, what is the best way to 36. is it trial and error? Is it wrong to use 144 to test this fact?

Thanks

EMPOWERgmatRichC · Answer

Hi Mo2men,

For Fact 2, the number 144 is a fine number to TEST (and it's relatively easy since you know that 12^2 is a multiple of 12). When it comes to TESTing VALUES, I'm always looking for the easiest numbers to TEST (and that often means looking for the 'smallest').

As far as how I ended up choosing 36, we know that P is an integer and that P^2 is a multiple of 12. So I was looking for a PERFECT SQUARE that was a multiple of 12. If you know your perfect squares, then choosing 36 isn't that big of a 'leap' in logic.

GMAT assassins aren't born, they're made,
Rich

Mo2men · Answer

Hi Rich,
Thanks for your prompt and easy logic.

Bunuel · Answer

MANHATTAN GMAT OFFICIAL SOLUTION:

The prime factorization of 96 is (2)(2)(2)(2)(2)(3) = (2^5)(3^1). In order for p^2 to be divisible by 96, p^2 would have to have the prime factors 2^5*3^1 in its prime box. The rephrased question is therefore “Does p^2 have at least five 2's and one 3 in its prime box?”

(1) INSUFFICIENT: If p is a multiple of 8 = (2)(2)(2), p has 2^3 in its prime box. Therefore, p^2 has (2^3)^2 = 2^6 in its prime box, and therefore has the required five 2's. However, it is uncertain whether p^2 has at least one 3 in its prime box.

Alternatively, we could list numbers: 
p = 0, 8, 16, 24, etc.
p^2 = 0, 64, 256, 576, etc.
Divisible by 96? Yes, No, No, Yes

(2) INSUFFICIENT: If p^2 is a multiple of 12 = (2)(2)(3), p^2 has two 2's and one 3 in its prime box. It is uncertain whether p^2 has at least five 2's total, as there may or may not be three more 2's in the prime box.

Alternatively, we could list numbers: 
p^2 = 0, 36, 144, 96^2, etc. (perfect squares that are multiples of 12)
Divisible by 96? Yes, No, No, Yes.

(1) AND (2) SUFFICIENT: We know from (1) that p^2 has 2^6 in its prime box, and we know from (2) that p^2 has a 3 in its prime box. Therefore, it is certain that p^2 has at least five 2's and one 3 in its prime box.

Note that the number listing approach would be a little cumbersome for the combined statements.

The correct answer is C.

anairamitch1804 · Answer

The prime factorization of 96 is (2)(2)(2)(2)(2)(3) = (25)(31). In order for p2 to be divisible by 96, p^2 would have to have the prime factors 2531 in its prime box. The rephrased question is therefore “Does p^2 have at least five 2's and one 3 in its prime box?”

(1) INSUFFICIENT: If p is a multiple of 8 = (2)(2)(2), p has 23 in its prime box. Therefore, p^2 has (2^3)^2 = 2^6 in its prime box, and therefore has the required five 2's. However, it is uncertain whether p^2 has at least one 3 in its prime box.
Alternatively, we could list numbers:
p = 0, 8, 16, 24, etc.
p^2 = 0, 64, 256, 576, etc.
Divisible by 96? Yes, No, No, Yes

(2) INSUFFICIENT: If p^2 is a multiple of 12 = (2)(2)(3), p^2 has two 2's and one 3 in its prime box. It is uncertain whether p^2 has at least five 2's total, as there may or may not be three more 2's in the prime box.
Alternatively, we could list numbers:
p^2 = 0, 36, 144, 962, etc. (perfect squares that are multiples of 12)
Divisible by 96? Yes, No, No, Yes
(1) AND (2) SUFFICIENT: We know from (1) that p^2 has 2^6 in its prime box, and we know from (2) that p^2 has a 3 in its prime box. Therefore, it is certain that p^2 has at least five 2's and one 3 in its prime box.

The correct answer is C.

Nunuboy1994 · Answer

Just gotta be careful with the distinction between the two statements

P= 3 x 2^5 x some integer k? let's find out

St 1

P=2^3 x some integer k clearly insufficient

St 2

P^2= 3 x 2^2 x some integer k

St 1 and St 2

Each of P must be divisible by 2^3 x some integer k so P^2 must be divisible by 2^6 which is enough 2's

C

BrentGMATPrepNow · Answer

Target question :    Is p² divisible by 96?  
This is a good candidate for  rephrasing the target question.

-----ASIDE---------------------
A lot of integer property questions can be solved using prime factorization. 
For questions involving divisibility, divisors, factors and multiples, we can say:
  If N is  divisible by k , then k is "hiding" within the prime factorization of N

Consider these examples: 
24 is divisible by  3  because 24 = (2)(2)(2) (3) 
Likewise, 70 is divisible by  5  because 70 = (2) (5) (7)
And 112 is divisible by  8  because 112 = (2) (2)  (2)  (2) (7)
And 630 is divisible by  15  because 630 = (2)(3) (3)  (5) (7)
--------------------------

Since 96 = (2)(2)(2)(2)(3), we can rephrase the target question as:
  REPHRASED target question :    Are there four 2's and one 3 hiding in the prime factorization of p² ?

Aside: the video below has tips on rephrasing the target question

Statement 1 : p is a multiple of 8  
In other words, p is divisible by 8
8 = (2)(2)(2) 
So, we know that there are at least three 2's hiding in the prime factorization of p
This also tells us that  there are SIX 2's hiding in the prime factorization of p² 
Unfortunately this information is not sufficient to answer the target question. 
Consider these two possible cases: 
 Case a : p = 8, in which case p² = 64. In this case, the answer to the target question is  NO, p² is NOT divisible by 96 
 Case b : p = 24, in which case p² = 576. In this case, the answer to the target question is  YES, p² IS divisible by 96 
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : p² is a multiple of 12 
12 = (2)(2)(3)
So, we know that there are at least two 2's and one 3 hiding in the prime factorization of p²
Unfortunately this information is not sufficient to answer the target question. 
Consider these two possible cases: 
 Case a : p = 12, in which case p² = 144 (which is divisible by 12). In this case, the answer to the target question is  NO, p² is NOT divisible by 96 
 Case b : p = 24, in which case p² = 576 (which is divisible by 12). In this case, the answer to the target question is  YES, p² IS divisible by 96 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
Statement 1 tells us that  there are SIX 2's hiding in the prime factorization of p² 
Statement 2 tells us that there is at least ONE 3 hiding in the prime factorization of p²
So, when we combine the two statements, we can be certain that  there are at least four 2's and one 3 hiding in the prime factorization of p² 
Since we can answer the  REPHRASED target question  with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent
RELATED VIDEO
 https://www.youtube.com/watch?v=MyG1K3ee69w

minustark · Answer

96 = 3 * 32 = 2^5 * 3. So, to be divisible by 96, p has to be expresses as 2^5* 3
1) p = 2^3 k . So, p = 2^6 *k^2. Depending on the value of k, p^2 will be divisible by 96 or not. not sufficient

2) p^2 = k*(2^2 * 3). Not sufficient.
Together, p^2 = 2^6 *3^2, it will be divisible by 96. Sufficient
C is the answer

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If p is a positive integer, is p^2 divisible by 96?

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GMAT Data Sufficiency (DS) Questions

If p is a positive integer, is p^2 divisible by 96?

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