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S is a set of n consecutive positive integers. Is the mean

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by BTGmoderatorDC » Tue Oct 30, 2018 5:32 pm

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Answers

A

B

C

D

E

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Difficulty—

S is a set of n consecutive positive integers. Is the mean of the set a positive integer?

(1) the range of S is an even integer

(2) the median of S is a positive integer

OA D

Source: Magoosh
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Source: — Data Sufficiency |
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by fskilnik@GMATH » Tue Oct 30, 2018 6:52 pm
BTGmoderatorDC wrote:S is a set of n consecutive positive integers. Is the mean of the set a positive integer?

(1) the range of S is an even integer

(2) the median of S is a positive integer

Source: Magoosh
$$S\,\,\,:\,\,\,n\,\,{\rm{consec}}{\rm{.}}\,\,{\rm{posit}}{\rm{.}}\,\,{\rm{ints}}\,\,\,\,\,\,\left( { \Rightarrow \,\,\,{\rm{mean}} > 0} \right)$$
$${\rm{mean}}\,\,\mathop = \limits^? \,\,{\rm{int}}\,\,\,\, \Leftrightarrow \,\,\,\,n\,\,\mathop = \limits^? \,\,{\rm{odd}}$$
$$\left( 1 \right)\,\,\max - \min = {\rm{even}}\,\,\,\, \Rightarrow \,\,\,\,n\,\,{\rm{odd}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( 2 \right)\,\,{\rm{mean}}\mathop = \limits^{\left( * \right)} {\rm{median}} = {\mathop{\rm int}} \,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( * \right)\,\,{\rm{finite}}\,\,{\rm{arithmetic}}\,\,{\rm{sequence}}$$

This solution follows the notations and rationale taught in the GMATH method.

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Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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by Jay@ManhattanReview » Tue Oct 30, 2018 9:24 pm
BTGmoderatorDC wrote:S is a set of n consecutive positive integers. Is the mean of the set a positive integer?

(1) the range of S is an even integer

(2) the median of S is a positive integer

OA D

Source: Magoosh
Say the set of set of n consecutive positive integers is

{x, (x + 1), (x + 2), (x + 3), ...}

Case 1: If n is odd, then the set is, say, {x, (x + 1), (x + 2)}. Mean = x + 1 = a positive integer.
Case 2: If n is even, then the set is, say, {x, (x + 1), (x + 2), (x + 3)}. Mean = [(x + 1) + (x + 2)]/2 = x + 1.5 = Not a positive integer.

So, the answer depends on whether n is odd or even.

Question rephrased: Is n odd?

Let's take each statement one by one.

(1) The range of S is an even integer.

Case 1: If n is odd, then the set is, say, {x, (x + 1), (x + 2)}. Mean = x + 1 = a positive integer. Range = x + 2 - x = 2 = an even integer
Case 2: If n is even, then the set is, say, {x, (x + 1), (x + 2), (x + 3)}. Mean = [(x + 1) + (x + 2)]/2 = x + 1.5 = Not a positive integer.
Range = x + 3 - x = 3 = Not an even integer.

Thus, Case 2 is not applicable/invalid. It implies that n is odd. Sufficient.

(2) The median of S is a positive integer.

Case 1: If n is odd, then the set is, say, {x, (x + 1), (x + 2)}. Mean = x + 1 = a positive integer. Median = x + 1 = a positive integer
Case 2: If n is even, then the set is, say, {x, (x + 1), (x + 2), (x + 3)}. Mean = [(x + 1) + (x + 2)]/2 = x + 1.5 = Not a positive integer.
Median = [(x + 1) + (x + 2)]/2 = x + 1.5 = Not a positive integer

Thus, Case 2 is not applicable/invalid. It implies that n is odd. Sufficient.

The correct answer: D

Hope this helps!

-Jay
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