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Can n /192 be an integer?

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by Jay@ManhattanReview » Thu Jun 08, 2017 7:00 am
jack0997 wrote:Can n /192 be an integer?

(1) n is a multiple of 24 but not 16.
(2) n is a multiple of 8 but not 48.

OA D
192 ƒ = 2^6 x 3
Thus, n/192 can be an integer only if n is divisible by 2^6 and by 3.
The exponent of '2' must be 6 or greater and exponent of '3' must be 1 or greater.

Statement 1:

n is a multiple of 24, i.e. 2^3 x 3 =ƒ> n is divisible by 2^3 and by 3.
n is not a multiple of 16, i.e. 2^4 ƒ=> n is not divisible by 2^4 (Maximum exponent of '2' is NOT 6.)
ƒ=> n is not divisible by 192.
Thus, n/192 cannot be an integer. - Sufficient

Statement 2:

n is a multiple of 8, i.e. 2^3 =ƒ> n is divisible by 2^3.
n is not a multiple of 48, i.e. 2^4 x 3.

Thus, there might be two possibilities:
1. Since n is not a multiple of 48, it is not a multiple 3; hence, it is definitely not a
multiple of 192.
2. The highest exponent of 2 by which n is divisible is 3, i.e. n is divisible by 2^3, but
not by 2^6, hence, it is definitely not a multiple of 192.

Thus, n/192 cannot be an integer. - Sufficient

The correct answer: D

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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by DavidG@VeritasPrep » Thu Jun 08, 2017 9:11 am
jack0997 wrote:Can n /192 be an integer?

(1) n is a multiple of 24 but not 16.
(2) n is a multiple of 8 but not 48.

OA D
Side note: a quick way to test divisibility is to break the number you're testing down to more manageable values.

Most of us can't eyeball 192 and see it's divisible by 16, but if we say 192 = 160 + 32, we can quickly see that both 160 and 32 are multiples of 16, and thus 192 is too. So if n is not divisible by 16 it can't be divisible by 192.

(It's also not obvious that 192 is a multiple of 48, but once we've established that 192 is a multiple of 16, and that 48 = 16*3, because we know that 192 is a multiple of 3, we know it must be a multiple of 16*3, or 48, as well.)
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by [email protected] » Thu Jun 08, 2017 12:51 pm
Hi jack 0997,

If you don't see a technical way to get to the correct answer, then a bit of 'brute force' arithmetic will work just as well. We ultimately have to define the 'patterns' involved in the two Facts that we're given.

We're asked if N/192 is an integer. This is a YES/NO question.

1) N is a multiple of 24 but not 16.

Let's list out the first few multiples of 24... 0, 24, 48, 72, 96, 120....
Now, which of these numbers is NOT a multiple of 16?...
0 IS a multiple of 16 (re: 0x16)
24 is NOT (re: 1x24)
48 IS a multiple of 16 (re: 3x16)
72 is NOT (re: 3x24)
96 is a multiple of 16 (re: 6x16)
120 is NOT (re: 5x24)
Etc.

From this information, you can derive several different patterns. Since we're focused on multiples of 24 that are NOT multiples of 16, we can see that all of the ODD multiples of 24 are what we're looking for. Is 192 an odd multiple of 24? How about 384 or any larger multiple?

192/24 = 8... which is NOT an odd multiple of 24
384/24 = 16... which is NOT an odd multiple of 24

Notice that we're only getting EVEN multiples of 24 here? You can continue to check other multiples if you like, but you'll find that the only values of N that 'fit' Fact 1 will ALWAYS produce a "NO" answer to the given question. That is a consistent result.
Fact 1 is SUFFICIENT.

2) N is a multiple of 8 but not 48.

We can do similar work with Fact 2, but I'm going to focus more on what N CANNOT be a multiple of... 48. Let's list out the first few multiples of 48... 0, 48, 96, 144, 192....

Notice that 192 is a multiple of 48. This means that multiples of 192 (re: 384, 576, 768, etc.) will ALSO be multiples of 48. Since Fact 2 tells us that N is NOT a multiple of 48, N/192 will NEVER be an integer... so the answer to the question is ALWAYS no.
Fact 2 is SUFFICIENT.

Final Answer: D

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by Matt@VeritasPrep » Thu Jun 08, 2017 5:25 pm
S1:

Since 192 is divisible by 16, any multiple of 192 must be a multiple of 16. n isn't a multiple of 16, so it can't be a multiple of 192: SUFFICIENT

S2:

Same idea. 192 is divisible by 48, so any multiple of 192 is a multiple of 48. n isn't a multiple of 48, so it can't be a multiple of 192: SUFFICIENT
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by Jay@ManhattanReview » Thu Jun 08, 2017 9:17 pm
jack0997 wrote:Can n /192 be an integer?

(1) n is a multiple of 24 but not 16.
(2) n is a multiple of 8 but not 48.

OA D
I see that in Statement 1, stating that 'n is a multiple of 24' is redundant. Mere stating that 'n is NOT a multiple of 16.' would suffice.

Since the statement 'n is NOT a multiple of 16.' means that n is not a multiple of 2^4, and for n to be a multiple of 192, it must be a multiple of 2^6; we need six 2s, however, 16 gives only four 2s, thus it's insufficient.

Similarly, in Statement 2, stating that 'n is a multiple of 8' is redundant. Mere stating that 'n is NOT a multiple of 48.' would suffice.

Since the statement 'n is NOT a multiple of 48.' means that n is not a multiple of 2^4, as with '16', it also does not give us six 2s; moreover it also does give us a 3 too (we need at least one 3 for n to be a multiple of 192), thus it's insufficient.

The correct answer: C

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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