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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S is a set of n consecutive positive integers. Is the mean
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BTGmoderatorDC
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$$S\,\,\,:\,\,\,n\,\,{\rm{consec}}{\rm{.}}\,\,{\rm{posit}}{\rm{.}}\,\,{\rm{ints}}\,\,\,\,\,\,\left( { \Rightarrow \,\,\,{\rm{mean}} > 0} \right)$$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
$${\rm{mean}}\,\,\mathop = \limits^? \,\,{\rm{int}}\,\,\,\, \Leftrightarrow \,\,\,\,n\,\,\mathop = \limits^? \,\,{\rm{odd}}$$
$$\left( 1 \right)\,\,\max - \min = {\rm{even}}\,\,\,\, \Rightarrow \,\,\,\,n\,\,{\rm{odd}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( 2 \right)\,\,{\rm{mean}}\mathop = \limits^{\left( * \right)} {\rm{median}} = {\mathop{\rm int}} \,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( * \right)\,\,{\rm{finite}}\,\,{\rm{arithmetic}}\,\,{\rm{sequence}}$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
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Say the set of set of n consecutive positive integers isBTGmoderatorDC 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
{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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