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# If x is a positive integer, which of the following must be divisible

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Math Expert
Joined: 02 Sep 2009
Posts: 51100
If x is a positive integer, which of the following must be divisible  [#permalink]

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22 May 2018, 03:28
00:00

Difficulty:

25% (medium)

Question Stats:

72% (01:10) correct 28% (01:11) wrong based on 62 sessions

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If x is a positive integer, which of the following must be divisible by 3?

A. (x+1)(x+2)(x+4)

B. (x)(x+2)(x+3)

C. (x)(x+3)(x+6)

D. (x−1)(x)(x+1)

E. (x)(x^2)(x^3)

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Re: If x is a positive integer, which of the following must be divisible  [#permalink]

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22 May 2018, 04:03

Solution

Given:
• The number x represents a positive integer

To find:
• Out of the given options, which one must be divisible by 3

Approach and Working:
• If we consider any 3 consecutive positive integers, one of them must be a multiple of 3
o Considering the options given, option D shows 3 consecutive integers (x – 1), x, and (x + 1)
• If one of them is a multiple of 3, then their product is also a definite multiple of 3
Hence, the correct answer is option D.

Note: with some examples, we can negate the other options.
• If x = 3, then (x + 1) (x + 2) (x + 4) = 4 * 5 * 7 [not a multiple of 3]
• If x = 2, then (x) (x + 2) (x + 3) = 2 * 4 * 5 [not a multiple of 3]
• If x = 1, then (x) (x + 3) (x + 6) = 1 * 4 * 7 [not a multiple of 3]
• If x = 2, then $$(x) (x^2) (x^3)$$ = 2 * 4 * 8 [not a multiple of 3]
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Re: If x is a positive integer, which of the following must be divisible  [#permalink]

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22 May 2018, 04:05
Option D is 3 consecutive integers. Since x is a positive integer, the minimum value of x is 1.
Therefore option D will always be divisible by 3.
Hence D = (x−1)(x)(x+1) is the answer.
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Re: If x is a positive integer, which of the following must be divisible &nbs [#permalink] 22 May 2018, 04:05
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