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Is the integer x divisible by 6?

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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Wed Nov 27, 2019 10:55 pm
BTGmoderatorDC wrote:Is the integer x divisible by 6?

(1) x + 3 is divisible by 3
(2) x + 3 is an odd number

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Source: GMAT Prep
Let's take each statement one by one.

(1) x + 3 is divisible by 3

Case 1: Say x = 6, then x + 3 = 9, divisible by 3. We see that x is divisible 6. The answer is yes.
Case 2: Say x = 9, then x + 3 = 12, divisible by 3. We see that x is not divisible 6. The answer is no.

No unique answer. Insufficient.

(2) x + 3 is an odd number.

Given that x + 3 is odd, x must be even. If x = 6, the answer is yes. However, if x = 8, the answer is no.

No unique answer. Insufficient.

(1) and (2) together

From (2), say x = 2n, where n is a positive integer. By taking x = 2n, we insured that x is even.

From (1), we know that (x + 3) = (2n + 3) is divisible by 3.

Or, 2n/3 + 3/3 = integer. Thus, 2n must be divisible by 3, or n must be divisible by 3. This implies that x = 2n is divisible by 2*3 = 6. Sufficient.

The correct answer: E

Hope this helps!

-Jay
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by swerve » Thu Nov 28, 2019 10:53 am
BTGmoderatorDC wrote:Is the integer x divisible by 6?

(1) x + 3 is divisible by 3
(2) x + 3 is an odd number

OA C

Source: GMAT Prep
1)\( x + 3\) is divisible by 3

\(x+3 = 3p\)
\(x = 3p - 3\)

\(p=-1, x=-6\)
\(p=0, x=-3\)
\(p=1, x=0\)
\(p=2, x=3\)
\(p=3, x=6\)
\(p=4, x=9\)
\(p=5, x=12\)

We can see that when \(p\) is odd, \(x\) will be even and divisible by 6.
Not sufficient because we don't know the value of \(p\) or \(x\). \(\Large{\color{red}\chi}\)

2) \(x + 3\) is an odd number
If \(x = 2\) then no, if \(x = 6\) then yes, in other words, we don't know if \(x\) is even or odd.
Not sufficient. \(\Large{\color{red}\chi}\)

Together, 2) \(x + 3\) is odd, so 1) \(x + 3 = 3p\) is odd, therefore \(3p\) is odd and \(x\) will be divisible by 6. \(\Large{\color{green}\checkmark}\)
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