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Intern  Joined: 14 Nov 2010
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If x is an integer, how many even numbers does set (0, x,  [#permalink]

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If x is an integer, how many even numbers does set (0, x, x^2, x^3,.... x^9) contain?

(1) The mean of the set is even
(2) The standard deviation of the set is 0

Originally posted by lhskev on 24 Dec 2010, 10:13.
Last edited by Bunuel on 12 Oct 2012, 09:42, edited 1 time in total.
Edited the question.
Math Expert V
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Re: How many even #'s does set X contain? Source: GMAT Club Test  [#permalink]

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lhskev wrote:
If X is an integer, how many even numbers does set (0, x, x^2, x^3,.... x^9) contain?

(1) The mean of the set is even
(2) The standard deviation of the set is 0

Thank you.

If x is an integer, how many even numbers does set (0, x, x^2, x^3,.... x^9) contain?

We have the set with 10 terms: {0, x, x^2, x^3, ..., x^9}.

Note that if $$x=odd$$ then the set will contain one even (0) and 9 odd terms (as if $$x=odd$$, then $$x^2=odd$$, $$x^3=odd$$, ..., $$x^9=odd$$) and if $$x=even$$ then the set will contain all even terms (as if $$x=even$$, then $$x^2=even$$, $$x^3=even$$, ..., $$x^9=even$$).

Also note that: standard deviation is always more than or equal to zero: $$SD\geq{0}$$. SD is 0 only when the list contains all identical elements (or which is same only 1 element).

(1) The mean of the set is even --> mean=sum/10=even --> sum=10*even=even --> 0+x+x^2+x^3+...+x^9=even --> x+x^2+x^3+...+x^9=even --> x=even (if x=odd then the sum of 9 odd numbers would be odd) --> all 10 terms in the set are even. Sufficient.

(2) The standard deviation of the set is 0 --> all 10 terms are identical --> as the first term is 0, then all other terms must equal to zero --> all 10 terms in the set are even. Sufficient.

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Manager  Joined: 25 Jun 2010
Posts: 72
Re: How many even #'s does set X contain? Source: GMAT Club Test  [#permalink]

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Statement 1:
Mean = Even
=> 0+x+x^2+x^3+...+x^9/10 = even

If X is odd, then 0+x+x^2+x^3+...+x^9 = 0 + Odd + Odd + .. + odd (total 9 times Odd) = Odd
So, x is Even,
There for all the members of the set are Even. ---- Sufficient

Statement 2:
The standard deviation of the set is 0 => All the members are 0 => All the members are Even

-- Sufficient

Manager  Joined: 27 Jul 2010
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Re: How many even #'s does set X contain? Source: GMAT Club Test  [#permalink]

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Nice explenation guys.
Manager  Joined: 05 Aug 2011
Posts: 62
Location: United States
Concentration: General Management, Sustainability
Re: How many even #'s does set X contain? Source: GMAT Club Test  [#permalink]

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Bunuel wrote:
lhskev wrote:
If X is an integer, how many even numbers does set (0, x, x^2, x^3,.... x^9) contain?

(1) The mean of the set is even
(2) The standard deviation of the set is 0

Thank you.

If x is an integer, how many even numbers does set (0, x, x^2, x^3,.... x^9) contain?

We have the set with 10 terms: {0, x, x^2, x^3, ..., x^9}.

Note that if $$x=odd$$ then the set will contain one even (0) and 9 odd terms (as if $$x=odd$$, then $$x^2=odd$$, $$x^3=odd$$, ..., $$x^9=odd$$) and if $$x=even$$ then the set will contain all even terms (as if $$x=even$$, then $$x^2=even$$, $$x^3=even$$, ..., $$x^9=even$$).

Also note that: standard deviation is always more than or equal to zero: $$SD\geq{0}$$. SD is 0 only when the list contains all identical elements (or which is same only 1 element).

(1) The mean of the set is even --> mean=sum/10=even --> sum=10*even=even --> 0+x+x^2+x^3+...+x^9=even --> x+x^2+x^3+...+x^9=even --> x=even (if x=odd then the sum of 9 odd numbers would be odd) --> all 10 terms in the set are even. Sufficient.

(2) The standard deviation of the set is 0 --> all 10 terms are identical --> as the first term is 0, then all other terms must equal to zero --> all 10 terms in the set are even. Sufficient.

I am a little confused about statement 1) especially if x is odd. For eg: consider x=3. Now if there are 3 elements in the set, they are (0,3,9). The mean of the set will be (0+3+9)/3=4 which is EVEN
Now consider 4 terms in this series with x=3 the set will be (0,3,9,27) then the mean would be (0+3+9+27)/4 = 39/4 which is not even. So x could be odd and still have the Mean of the set to be even. So A is insufficient. What am I missing, thanks
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Re: How many even #'s does set X contain? Source: GMAT Club Test  [#permalink]

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ace312 wrote:
Bunuel wrote:
lhskev wrote:
If X is an integer, how many even numbers does set (0, x, x^2, x^3,.... x^9) contain?

(1) The mean of the set is even
(2) The standard deviation of the set is 0

Thank you.

If x is an integer, how many even numbers does set (0, x, x^2, x^3,.... x^9) contain?

We have the set with 10 terms: {0, x, x^2, x^3, ..., x^9}.

Note that if $$x=odd$$ then the set will contain one even (0) and 9 odd terms (as if $$x=odd$$, then $$x^2=odd$$, $$x^3=odd$$, ..., $$x^9=odd$$) and if $$x=even$$ then the set will contain all even terms (as if $$x=even$$, then $$x^2=even$$, $$x^3=even$$, ..., $$x^9=even$$).

Also note that: standard deviation is always more than or equal to zero: $$SD\geq{0}$$. SD is 0 only when the list contains all identical elements (or which is same only 1 element).

(1) The mean of the set is even --> mean=sum/10=even --> sum=10*even=even --> 0+x+x^2+x^3+...+x^9=even --> x+x^2+x^3+...+x^9=even --> x=even (if x=odd then the sum of 9 odd numbers would be odd) --> all 10 terms in the set are even. Sufficient.

(2) The standard deviation of the set is 0 --> all 10 terms are identical --> as the first term is 0, then all other terms must equal to zero --> all 10 terms in the set are even. Sufficient.

I am a little confused about statement 1) especially if x is odd. For eg: consider x=3. Now if there are 3 elements in the set, they are (0,3,9). The mean of the set will be (0+3+9)/3=4 which is EVEN
Now consider 4 terms in this series with x=3 the set will be (0,3,9,27) then the mean would be (0+3+9+27)/4 = 39/4 which is not even. So x could be odd and still have the Mean of the set to be even. So A is insufficient. What am I missing, thanks

consider x=3. Now if there are 3 elements in the set - You cannot have 3 elements. The set must contain $$0, 3, 3^2,3^3,3^4,...,3^9$$ - $$10$$ elements.
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Re: How many even #'s does set X contain? Source: GMAT Club Test  [#permalink]

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Can you please explain why we are considering 0 as even, because 0 is mostly treated as nether even nor odd. Hence 2) is insufficient.
Math Expert V
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Re: How many even #'s does set X contain? Source: GMAT Club Test  [#permalink]

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ssingla wrote:
Can you please explain why we are considering 0 as even, because 0 is mostly treated as nether even nor odd. Hence 2) is insufficient.

Zero is an even integer. Zero is nether positive nor negative, but zero is definitely an even number.

An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even (in fact zero is divisible by every integer except zero itself).

For more check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it helps.
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Re: If x is an integer, how many even numbers does set (0, x,  [#permalink]

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1
Tricky Question ...

nice explanation bunuel...

I didnt c zero in set..selected option E.. _________________
Bole So Nehal.. Sat Siri Akal.. Waheguru ji help me to get 700+ score !
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Re: If x is an integer, how many even numbers does set (0, x,  [#permalink]

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Bumping for review and further discussion.
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Re: If x is an integer, how many even numbers does set (0, x,  [#permalink]

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For Stmt A ?

Does it mean if x = odd , then sum is odd , the mean would be = Sum / numbers ( 9 ) in this case which would be

Mean = Sum ( Odd )
------------------ == Odd .. so X can only be even here ...
9 ( Odd )
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Re: If x is an integer, how many even numbers does set (0, x,  [#permalink]

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