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Math Expert V
Joined: 02 Sep 2009
Posts: 56300
How many numbers of 7 consecutive positive integers are divisible by 6  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 63% (01:53) correct 37% (02:05) wrong based on 85 sessions

### HideShow timer Statistics How many numbers of 7 consecutive positive integers are divisible by 6?

(1) Their average is divisible by 6
(2) Their median is divisible by 12

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Math Expert V
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Posts: 7764
Re: How many numbers of 7 consecutive positive integers are divisible by 6  [#permalink]

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Bunuel wrote:
How many numbers of 7 consecutive positive integers are divisible by 6?

(1) Their average is divisible by 6
(2) Their median is divisible by 12

Hi,

the answer lies in NUMBER PROPERTIES..

7 consecutive positive number will have either 1 or 2 numbers div by 6..
If we know first of these or the last of these is div by 6, there are 2 numbers then- 1st and 7th
otherwise ONLY 1..

lets check the choices:-
(1) Their average is divisible by 6
Average will be equal to the middle number, so 1st is not div by 6..
ONLY 1 is div
Suff

(2) Their median is divisible by 12
Same as statement 1..
average = median for an AP
Suff

D
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Re: How many numbers of 7 consecutive positive integers are divisible by 6  [#permalink]

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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

How many numbers of 7 consecutive positive integers are divisible by 6?

(1) Their average is divisible by 6
(2) Their median is divisible by 12

In the original condition, there is 1 variable since there are 7 consecutive positive integers. In order to match with the number of equations, you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.
For 1), there is always 1, which is unique and sufficient.
For 2), there is always 1 as well, which is unique and sufficient.

 For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: How many numbers of 7 consecutive positive integers are divisible by 6  [#permalink]

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1
Hi All,

We're told that we have 7 CONSECUTIVE positive integers. We're asked how many of them are divisible by 6. To start, with 7 consecutive integers, only 1 or 2 of those integers will be divisible by 6. Once you have a number that is divisible by 6, the 'next' number that is divisible by 6 will either be '6 more' or '6 less' than the original number. For example: 0, 6, 12, 18, 24, etc.

1) Their average is divisible by 6.

With 7 consecutive integers, the AVERAGE will equal the "middle" number (in this case, the 4th number), so we can use a bit of 'brute force' to determine whether a pattern exists here or not.
IF... the numbers are...
3, 4, 5, 6, 7, 8, 9 .... there's 1 number that is divisible by 6.
9, 10, 11, 12, 13, 14, 15 .... there's 1 number that is divisible by 6.
15, 16, 17, 18, 19, 20, 21 .... there's 1 number that is divisible by 6.
Etc.
There will ALWAYS be just 1 number that is divisible by 6.
Fact 1 is SUFFICIENT

2) Their median is divisible by 12

With 7 consecutive integers, the AVERAGE = MEDIAN and both values will equal the "middle" number (in this case, the 4th number), so we can use a bit of 'brute force' here too. The difference is that the 4th value will have to be a multiple of 12.
IF... the numbers are....
9, 10, 11, 12, 13, 14, 15 .... there's 1 number that is divisible by 6.
21, 22, 23, 24, 25, 26, 27 .... there's 1 number that is divisible by 6.
Etc.
There will ALWAYS be just 1 number that is divisible by 6.
Fact 2 is SUFFICIENT

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Rich
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Re: How many numbers of 7 consecutive positive integers are divisible by 6  [#permalink]

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_________________ Re: How many numbers of 7 consecutive positive integers are divisible by 6   [#permalink] 27 Nov 2018, 07:52
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