If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12?
(1) x^2 + 2x is a multiple of 3.
(2) 3x is a multiple of 2.
OA B
Source: Manhattan Prep
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If x is a positive integer, is (x)(x + 2)(x + 4) divisible
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BTGmoderatorDC
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Let's take each statement one by one.BTGmoderatorDC wrote:If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12?
(1) x^2 + 2x is a multiple of 3.
(2) 3x is a multiple of 2.
OA B
Source: Manhattan Prep
(1) x^2 + 2x is a multiple of 3.
=> x( x + 2) is a multiple of 3.
Thus, for (x)(x + 2)(x + 4) to be divisible by 12, (x + 4) must be divisible by 4, which is not known. Insufficient.
(2) 3x is a multiple of 2.
=> x is even
Say x = 2n, where n is a positive integer.
Thus, (x)(x + 2)(x + 4) = (2n)(2n + 2)(2n + 4) = 8n(n + 1)(n + 2)
We see that n(n + 1)(n + 2) is a product of three consecutive positive integers; thus, n(n + 1)(n + 2) must be divisible by 8*3 = 24.
Thus, 8n(n + 1)(n + 2) = (x)(x + 2)(x + 4) is divisible by 24 or 12. Sufficient.
The correct answer: B
Hope this helps!
-Jay
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