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# Is the positive integer n a multiple of 24 ?

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Re: 115. Multiple [#permalink]
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i'd say E..

12 is a multiple of both 4 and 6 but not of 24.
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Re: 115. Multiple [#permalink]
dreambeliever wrote:
i'd say E..

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

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.
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Re: 115. Multiple [#permalink]
fluke wrote:
dreambeliever wrote:
i'd say E..

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

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.

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.
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Re: 115. Multiple [#permalink]
Baten80 wrote:
fluke wrote:
dreambeliever wrote:
i'd say E..

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

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.

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.

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 ?
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Re: Is the positive integer n a multiple of 24 ? [#permalink]
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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

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Is the positive integer n a multiple of 24 ? [#permalink]
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Bunuel wrote:
Is the positive integer n a multiple of 24 ?

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

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

Originally posted by BrentGMATPrepNow on 25 Apr 2020, 07:51.
Last edited by BrentGMATPrepNow on 17 May 2021, 09:46, edited 1 time in total.
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Is the positive integer n a multiple of 24 ? [#permalink]
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MHIKER wrote:
Is the positive integer n a multiple of 24?

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

(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 $$E$$
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Re: Is the positive integer n a multiple of 24 ? [#permalink]
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.
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Re: Is the positive integer n a multiple of 24 ? [#permalink]
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Sneha2021 wrote:
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.

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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Re: Is the positive integer n a multiple of 24 ? [#permalink]
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Re: Is the positive integer n a multiple of 24 ? [#permalink]
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