Is the positive integer n a multiple of 24 ?

Question

Is the positive integer n a multiple of 24?

(1) n is a multiple of 4. 
(2) n is a multiple of 6.

Bunuel · Accepted Answer

There is no difference between saying that 12 is a multiple of 4 and that 12 is divisible by 4.

As for the question.
 Is the positive integer n a multiple of 24? 
 
(1) n is a multiple of 4. Not sufficient.
(2) n is a multiple of 6. Not sufficient.

(1)+(2) n is a multiple of both 4 and 6 which means that  it's a multiple of least common multiple of 4 and 6 , which is 12. So, even taken together statements are not sufficient, since n can be for example 12 as well as 24. Not sufficient.

Answer: E.

Generally if a positive integer  n  is a multiple of positive integer  a  and positive integer  b , then  n  is a multiple of LCM(a,b).

Hope it helps.

WoundedTiger · Answer

Is the positive integer n a multiple of 24 ?

(1) n is a multiple of 4.
(2) n is a multiple of 6.

Sol: Is n= 24I where I is an Integer

St 1: n=4a where "a" is an integer. Now If n=4 then no but n=48 then yes.

A and D are ruled out

St2: If n=6 then no if n=24 then yes.

Combining we get n is muliple of 4 and 6 so n is a mulitple of 12 but need not be a multiple of 24...here's why

If n=12,36,60 then not a mutiple of 24, but
If n= 24,48 then Yes

Ans is E

dreambeliever · Answer

i'd say E..

12 is a multiple of both 4 and 6 but not of 24.

fluke · Answer

Yes, you are right. I ignored the fact that the 2 in the prime factor of 6 may be the same 2 from the prime factor of 2's in the factors of 12. Thus, n definitely has only two 2's and one 3 as factor, which is 12. thanks.

MHIKER · Answer

fluke please make clear in your way such as:

Prime factors of 24: 2^3*3
(1) 4: 2^2; Not sufficient.
(2) 6: 2*3; Not sufficient.

Combining both; minimum factors of n= 2^2*2*3 = 2^3*3 = all factors of 24. Sufficient.

siddhans · Answer

If in the same question we were to replace 'multiple' by 'divisible' what the difference??? What exactly happens when something is a multiple of something or when something is a divisbile of something ?

b2bt · Answer

A) If n is a positive multiple of 4 it should be of the form 4*x where x is 1, 2, 3,4 etc
therefore, n can be 4, 8 , 24

NOT SUFFICIENT

B) If n is a positive multiple of 6 it should be of the form 6*x where x is 1, 2, 3,4 etc
therefore, n can be 6,12,18,24, etc

NOT SUFFICIENT

Both A&B ) n is multiple of both 4 and 6 i.e. it is of the form 4*6*x = 24*x

SUFFICIENT

Difficulty level - 500
Time Taken - 00:17

BrentGMATPrepNow · Answer

Target question :    Is the positive integer n a multiple of 24 ?  
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  a multiple of k , then k is "hiding" within the prime factorization of N

Consider these examples: 
24 is a multiple of  3  because 24 = (2)(2)(2) (3) 
Likewise, 70 is a multiple of  5  because 70 = (2) (5) (7)
And 112 is a multiple of  8  because 112 = (2) (2)  (2)  (2) (7)
And 630 is a multiple of  15  because 630 = (2)(3) (3)  (5) (7)
-----ONTO THE QUESTION!---------------------

Since 24 = (2)(2)(2)(3), we can rephrase the target question as follows...
  REPHRASED target question :    Are three 2's and one 3 hiding in the prime factorization of n?

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

Statement 1 : n is a multiple of 4.  
4 = (2)(2), so statement 1 is telling us that there are two 2's hiding in the prime factorization of n. 
Of course, there COULD be additional 2's (and 3's for that matter) hiding in the prime factorization of n.
Given this, there's no way we can answer the target question with certainty. 
If you're not convinced, consider these two possible cases:
 Case a : n = 12 (which is a multiple of 4). In this case, the answer to the target question is  NO, n is not divisible by 24 
 Case b : n = 48 (which is a multiple of 4). In this case, the answer to the target question is  YES, n is divisible by 24 
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : n is a multiple of 6 
6 = (2)(3), so statement 2 is telling us that there is one 2 and one 3 hiding in the prime factorization of n. 
Of course, there COULD be additional 2's and 3's for that matter hiding in the prime factorization of n.
Given this, there's no way we can answer the target question with certainty. 
Consider these two possible cases:
 Case a : n = 12 (which is a multiple of 6). In this case, the answer to the target question is  NO, n is not divisible by 24 
 Case b : n = 48 (which is a multiple of 6). In this case, the answer to the target question is  YES, n is divisible by 24 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
IMPORTANT: Notice that I was able to use the  same counter-examples  to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED. 
In other words, 
 Case a : n = 12 (which is a multiple of 4 and 6). In this case, the answer to the target question is  NO, n is not divisible by 24 
 Case b : n = 48 (which is a multiple of 4 and 6). In this case, the answer to the target question is  YES, n is divisible by 24 
Since we cannot answer the  target question  with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

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

MHIKER · Answer

(1) Let,  n=4,8,12, \ or \ 24  then the answer will be Yes and No. Insufficient.

(2) Let,  n=6,12,18, or \ 24  then the answer will be Yes and No. Insufficient.

Considering both:

n=12  or  24  Still answer is NO or YES. Insufficient.

The answer is  E

Sneha2021 · Answer

Bunuel 
If the question says "n" can be any integer, then answer would still be E

(1) Let, n=-4,0, 4,8,12, or 24

(2) Let, n=-6, 0, 6,12,18,or 24

OR n=-12,0,12

Please let me know if this is correct.

Bunuel · Answer

If the answer is E when there are more constraints, then it would still be E when there are fewer constraints. Therefore, if we were not informed that n is a positive integer, the answer would remain E.

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Is the positive integer n a multiple of 24 ?

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