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Is n/12 an integer?

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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Thu Jun 08, 2017 6:14 am
jack0997 wrote:Is n/12 an integer?

(1) n^2/144 is an integer.
(2) n/6 is an integer.

OA: C
Statement 1:

We know that n^2/144 is an integer, i.e. n^2 is divisible by 144. However, it is not known whether n is an integer.

We may have,

Case 1: Say n = √288 ƒ=> n^2 ƒ = 288, which is divisible by 144.
However, n is not divisible by 12. - The answer is 'No.'

Case 2: Say n ƒ= 24 ƒ=> n^2 ƒ = 24^2, which is divisible by 144.
Also, n is divisible by 12. - The answer is 'Yes.'

Thus, there is no unique answer. - Insufficient

Statement 2:

We know that n/6 is an integer, i.e. n is divisible by 6, hence n must be an integer.

We may have:

Case 1: n ƒ= 18, which is divisible by 6.
However, n is not divisible by 12. - The answer is 'No.'
Case 2: n =ƒ 24, which is divisible by 6.
Also, n is divisible by 12. - The answer is 'Yes.'

Thus, there is no unique answer. - Insufficient

Statement 1 &2 together:

From the Statement 2, we know that n is an integer.
Thus, from the first statement, we have: n^2 is an integer, which is divisible by 144 ƒ = 12^2
ƒ=> n is divisible by 12. - Sufficient

The correct answer: C

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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by Matt@VeritasPrep » Thu Jun 08, 2017 5:29 pm
S1:

n²/144 = integer

(n/12)² = integer

n/12 = √integer

n = 12*√integer

This is pretty close to what we want, but we don't know if √integer is itself an integer! For instance, we could have n² = 288, in which case n²/144 = 2, but n = 12√2. Not sufficient!

S2:

n/6 = integer

n = 6 * integer

This tells us that n is an integer, but it doesn't HAVE to be a multiple of 12. Not sufficient!

Together, S2 tells us that n is an integer, and S1 tells us that IF n is an integer, it's divisible by 12. So with the two statements, we get n = integer, then n = a multiple of 12.
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by Jay@ManhattanReview » Thu Jun 08, 2017 9:01 pm
jack0997 wrote:Is n/12 an integer?

(1) n^2/144 is an integer.
(2) n/6 is an integer.

OA: C
Let's take an Algebraic route to this problem.

Statement 1: n^2/144 is an integer

Say n^2 = 144k; where k is an integer

=> n = 12√k
=> n/12 = √k

If √k is an integer, the answer is Yes, else No. No unique answer. Insufficient.

Statement 2: n/6 is an integer

Say n = 6m; where m is an integer

=> n/12 = m/2

If m/2 is an integer, the answer is Yes, else No. No unique answer. Insufficient.

Statement 1 & 2 together:

From both the statements, we get

√k = m/2

=> m = 2√k

Since m is an integer, √k must be an integer. Thus, n/12 = √k = Integer. Sufficient.

The correct answer: C

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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