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# S is a set of positive integers. The average of the terms in

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S is a set of positive integers. The average of the terms in [#permalink]

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23 Sep 2012, 20:27
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S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is the sum of all the integers in S?

(1) The range of S is a prime number that is less than 11 and is not a factor of 10.
(2) S is composed of 5 different integers.
[Reveal] Spoiler: OA

Last edited by Bunuel on 07 Nov 2013, 11:43, edited 2 times in total.
Renamed the topic.

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23 Sep 2012, 23:15
rohitgarg wrote:
S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is
the sum of all the integers in S?
(1) The range of S is a prime number that is less than 11 and is not a factor of 10.
(2) S is composed of 5 different integers.

Based on option A - prime numbers less than 11 are 2, 3, 5, 7. 2 & 5 is rejected because they are factors of 10. we get two numbers - 3 & 7.- Insuff.
Based on option B - multiple combination possible - Insuff.

Together - 3 & 7 both doesn't fall under creteria. Answer E.

Cheers!
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24 Sep 2012, 00:18
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rohitgarg wrote:
S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is
the sum of all the integers in S?
(1) The range of S is a prime number that is less than 11 and is not a factor of 10.
(2) S is composed of 5 different integers.

A - average
(1) Primes less than 11 are 2, 3, 5, and 7. Not factor of 10, we are left with 3 and 7.
If A = 3, we can have {2, 2, 3, 5} or {1, 2, 3, 4, 4, 4}.
Not sufficient.

(2) For 5 evenly spaced numbers, k - 2d, k - d, k, k + d, k + 2d the range is 4d and the average is k.
We can simply take k = 4d, and we have infinitely many sets fulfilling the condition of the form {2d, 3d, 4d, 5d, 6d}.
For example {2, 3, 4, 5, 6}, {20, 30, 40, 50, 60}...
Not sufficient.

(1) and (2) together:
We have seen above that the range can be either 3 or 7.
If the range is 3, we cannot have 5 distinct integers in the set, so only 7 is left.

We know that the sum of the 5 integers is 5 * 7 = 35, the smallest number is k and the largest number is k + 7, where k is some positive integer.
Necessarily 35 - 7 = 28 > 5k, so k must be not greater than 5.
If k = 5, the first and the last number are 5 and 12. The smallest 3 remaining integers are 6, 7, and 8 together with 5 and 12 will give a sum of 37.
If k = 4, we find sets which fulfill the condition: {4, 5, 7, 8, 11} and also (4, 5, 6, 9, 11}.
Obviously, not sufficient.

CORRECTION:

What a miss! We have to find the sum of the numbers and not the set of numbers. Although there is more than one possibility, the total sum is clearly 7*5 = 35.

And IanStewart is right about (1), in a set of numbers, each element is different.
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Last edited by EvaJager on 24 Sep 2012, 22:55, edited 2 times in total.

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24 Sep 2012, 01:17
EvaJager wrote:
rohitgarg wrote:
S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is
the sum of all the integers in S?
(1) The range of S is a prime number that is less than 11 and is not a factor of 10.
(2) S is composed of 5 different integers.

A - average
(1) Primes less than 11 are 2, 3, 5, and 7. Not factor of 10, we are left with 3 and 7.
If A = 3, we can have {2, 2, 3, 5} or {1, 2, 3, 4, 4, 4}.
Not sufficient.

(2) For 5 evenly spaced numbers, k - 2d, k - d, k, k + d, k + 2d the range is 4d and the average is k.
We can simply take k = 4d, and we have infinitely many sets fulfilling the condition of the form {2d, 3d, 4d, 5d, 6d}.
For example {2, 3, 4, 5, 6}, {20, 30, 40, 50, 60}...
Not sufficient.

(1) and (2) together:
We have seen above that the range can be either 3 or 7.
If the range is 3, we cannot have 5 distinct integers in the set, so only 7 is left.

We know that the sum of the 5 integers is 5 * 7 = 35, the smallest number is k and the largest number is k + 7, where k is some positive integer.
Necessarily 35 - 7 = 28 > 5k, so k must be not greater than 5.
If k = 5, the first and the last number are 5 and 12. The smallest 3 remaining integers are 6, 7, and 8 together with 5 and 12 will give a sum of 37.
If k = 4, we find sets which fulfills the condition: {4, 5, 7, 8, 11} and also (4, 5, 6, 9, 11}.
Obviously, not sufficient.

@EvaJager - in (2) why are you considering "5 evenly spaced numbers"? This is not mentioned anywhere in the question.
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24 Sep 2012, 03:45
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Capricorn369 wrote:
EvaJager wrote:
rohitgarg wrote:
S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is
the sum of all the integers in S?
(1) The range of S is a prime number that is less than 11 and is not a factor of 10.
(2) S is composed of 5 different integers.

A - average
(1) Primes less than 11 are 2, 3, 5, and 7. Not factor of 10, we are left with 3 and 7.
If A = 3, we can have {2, 2, 3, 5} or {1, 2, 3, 4, 4, 4}.
Not sufficient.

(2) For 5 evenly spaced numbers, k - 2d, k - d, k, k + d, k + 2d the range is 4d and the average is k.
We can simply take k = 4d, and we have infinitely many sets fulfilling the condition of the form {2d, 3d, 4d, 5d, 6d}.
For example {2, 3, 4, 5, 6}, {20, 30, 40, 50, 60}...
Not sufficient.

(1) and (2) together:
We have seen above that the range can be either 3 or 7.
If the range is 3, we cannot have 5 distinct integers in the set, so only 7 is left.

We know that the sum of the 5 integers is 5 * 7 = 35, the smallest number is k and the largest number is k + 7, where k is some positive integer.
Necessarily 35 - 7 = 28 > 5k, so k must be not greater than 5.
If k = 5, the first and the last number are 5 and 12. The smallest 3 remaining integers are 6, 7, and 8 together with 5 and 12 will give a sum of 37.
If k = 4, we find sets which fulfill the condition: {4, 5, 7, 8, 11} and also (4, 5, 6, 9, 11}.
Obviously, not sufficient.

@EvaJager - in (2) why are you considering "5 evenly spaced numbers"? This is not mentioned anywhere in the question.

It is the easiest to express the average when the numbers are evenly spaced. The goal is to find as easily as possible one/more examples which fulfill the condition.
And it is nowhere stated that the numbers cannot be evenly spaced. Also, in (1) it was not mentioned that some of the numbers cannot be equal.
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Re: S is a set of positive integers. The average of the terms in [#permalink]

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24 Sep 2012, 22:37
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rohitgarg wrote:
S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is the sum of all the integers in S?

(1) The range of S is a prime number that is less than 11 and is not a factor of 10.
(2) S is composed of 5 different integers.

Neither statement is sufficient alone.

From Statement 1, the range (and thus the average) is either 3 or 7. From Statement 2, we know that the range cannot be 3, since no set with a range of 3 could contain five different integers. So the mean is 7, and we have 5 things in our set, so the sum is 5*7=35 which is what the question asked for. The answer is C.

If the OA in the original source is 'E' here because the OA claims that S could contain repeated elements, the OA is wrong. The ambiguous wording of Statement 2 aside, the word 'set' has a precise definition in mathematics, and sets cannot contain repeated elements. You don't need to know that for the GMAT, but there's a reason GMAT questions always talk about 'lists of values' or 'data sets' or use some similar term in questions where repeated elements might be relevant.
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25 Oct 2012, 15:27
EvaJager wrote:
If k = 5, the first and the last number are 5 and 12. The smallest 3 remaining integers are 6, 7, and 8 together with 5 and 12 will give a sum of 37.
If k = 4, we find sets which fulfill the condition: {4, 5, 7, 8, 11} and also (4, 5, 6, 9, 11}.
Obviously, not sufficient.

^^^ what's this that you have done , can you please explain thinking why did you calculate the above, I would like to mention that I understand the problem sufficiently clear.

Thanks..

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26 Oct 2012, 07:26
himanshuhpr wrote:
EvaJager wrote:
If k = 5, the first and the last number are 5 and 12. The smallest 3 remaining integers are 6, 7, and 8 together with 5 and 12 will give a sum of 37.
If k = 4, we find sets which fulfill the condition: {4, 5, 7, 8, 11} and also (4, 5, 6, 9, 11}.
Obviously, not sufficient.

^^^ what's this that you have done , can you please explain thinking why did you calculate the above, I would like to mention that I understand the problem sufficiently clear.

Thanks..

I edited my first post, see the CORRECTION attached :

What a miss! We have to find the sum of the numbers and not the set of numbers. Although there is more than one possibility, the total sum is clearly 7*5 = 35.

And IanStewart is right about (1), in a set of numbers, each element is different.

I was trying to find sets which fulfill the given conditions, which was in fact unnecessary. We were only asked what is the sum of the numbers.
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27 Oct 2012, 12:35
EvaJager wrote:
himanshuhpr wrote:
EvaJager wrote:
If k = 5, the first and the last number are 5 and 12. The smallest 3 remaining integers are 6, 7, and 8 together with 5 and 12 will give a sum of 37.
If k = 4, we find sets which fulfill the condition: {4, 5, 7, 8, 11} and also (4, 5, 6, 9, 11}.
Obviously, not sufficient.

^^^ what's this that you have done , can you please explain thinking why did you calculate the above, I would like to mention that I understand the problem sufficiently clear.

Thanks..

I edited my first post, see the CORRECTION attached :

What a miss! We have to find the sum of the numbers and not the set of numbers. Although there is more than one possibility, the total sum is clearly 7*5 = 35.

And IanStewart is right about (1), in a set of numbers, each element is different.

I was trying to find sets which fulfill the given conditions, which was in fact unnecessary. We were only asked what is the sum of the numbers.

EvaJager.... could you please explain the entire solution.....please do it independently and not as an extension of some other post...

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27 Oct 2012, 12:48
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EvaJager.... could you please explain the entire solution.....please do it independently and not as an extension of some other post...[/quote]

This is my original post:

s-is-a-set-of-positive-integers-the-average-of-the-terms-in-139444.html#p1124575

Which part didn't you understand?

As I mentioned before, I misunderstood the question and I was trying to find concrete sets of numbers which fulfill the conditions.

Please, refer to IanStewart's post for a correct solution:

s-is-a-set-of-positive-integers-the-average-of-the-terms-in-139444.html#p1124943
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Re: S is a set of positive integers. The average of the terms in [#permalink]

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07 Nov 2013, 10:46
could you please edit the OA..??? since the Answer is C. it still shows E!~

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Re: S is a set of positive integers. The average of the terms in [#permalink]

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07 Nov 2013, 11:43
Supernova29 wrote:
could you please edit the OA..??? since the Answer is C. it still shows E!~

Edited. Thank you.
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Re: S is a set of positive integers. The average of the terms in [#permalink]

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09 Nov 2013, 03:30
Thanks Bunuel...

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Re: S is a set of positive integers. The average of the terms in [#permalink]

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20 Dec 2013, 04:36
(A+B+C+D+E)/5=E-A
A+B+C+D+E= 5(E-A)

Range can only by a 3 or a 7: 5*3=15

Therefore, the sum of A+B+C+D+E =15
this can only happen if A+B+C+D+E = 1+2+3+4+5

n(n+1)/2=(5*6)/2=15

but it cannot be 15 because 5-1 is 4 which is not a prime number. Therefore, the range must be 7 and the sum equal to 35.

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21 Dec 2013, 00:27
Capricorn369 wrote:
EvaJager wrote:
rohitgarg wrote:
S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is
the sum of all the integers in S?
(1) The range of S is a prime number that is less than 11 and is not a factor of 10.
(2) S is composed of 5 different integers.

A - average
(1) Primes less than 11 are 2, 3, 5, and 7. Not factor of 10, we are left with 3 and 7.
If A = 3, we can have {2, 2, 3, 5} or {1, 2, 3, 4, 4, 4}.
Not sufficient.

(2) For 5 evenly spaced numbers, k - 2d, k - d, k, k + d, k + 2d the range is 4d and the average is k.
We can simply take k = 4d, and we have infinitely many sets fulfilling the condition of the form {2d, 3d, 4d, 5d, 6d}.
For example {2, 3, 4, 5, 6}, {20, 30, 40, 50, 60}...
Not sufficient.

(1) and (2) together:
We have seen above that the range can be either 3 or 7.
If the range is 3, we cannot have 5 distinct integers in the set, so only 7 is left.

We know that the sum of the 5 integers is 5 * 7 = 35, the smallest number is k and the largest number is k + 7, where k is some positive integer.
Necessarily 35 - 7 = 28 > 5k, so k must be not greater than 5.
If k = 5, the first and the last number are 5 and 12. The smallest 3 remaining integers are 6, 7, and 8 together with 5 and 12 will give a sum of 37.
If k = 4, we find sets which fulfills the condition: {4, 5, 7, 8, 11} and also (4, 5, 6, 9, 11}.
Obviously, not sufficient.

@EvaJager - in (2) why are you considering "5 evenly spaced numbers"? This is not mentioned anywhere in the question.

From statement 1 and statement 2, we have range = 7 ( cannot be 2,5 or 3), hence average is also 7, hence sum = 5*7 = 35

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Re: S is a set of positive integers. The average of the terms in [#permalink]

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18 May 2015, 06:38
I think the answer should still be E.

Statement two says that set S is composed of 5 different integers- this IMO doesn't mean that S is composed of exactly 5 integers.

Two contrasting examples:

S= {3,6,7,9,10) - range = 7 and mean = 7. The Sum is 35
S= {3,6,7,7,9,10)- range = 7 and mean = 7. The Sum is 42

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Re: S is a set of positive integers. The average of the terms in [#permalink]

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18 May 2015, 06:41
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vivin0292 wrote:
I think the answer should still be E.

Statement two says that set S is composed of 5 different integers- this IMO doesn't mean that S is composed of exactly 5 integers.

Two contrasting examples:

S= {3,6,7,9,10) - range = 7 and mean = 7. The Sum is 35
S= {3,6,7,7,9,10)- range = 7 and mean = 7. The Sum is 42

(2) says that S is composed of 5 different integers.
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Re: S is a set of positive integers. The average of the terms in [#permalink]

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Re: S is a set of positive integers. The average of the terms in [#permalink]

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05 Jan 2017, 06:02
Quote:
It is the easiest to express the average when the numbers are evenly spaced. The goal is to find as easily as possible one/more examples which fulfill the condition.
And it is nowhere stated that the numbers cannot be evenly spaced. Also, in (1) it was not mentioned that some of the numbers cannot be equal.

Nowhere stated that the numbers cannot be evenly spaced DOESN'T MEAN that the numbers are evenly spaced. Do you have another explanation?

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Re: S is a set of positive integers. The average of the terms in [#permalink]

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06 Jan 2017, 11:12
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rohitgarg wrote:
S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is the sum of all the integers in S?

(1) The range of S is a prime number that is less than 11 and is not a factor of 10.
(2) S is composed of 5 different integers.

We are given that S is a set of positive integers and the average of the terms in S is equal to the range of the terms in S. We need to determine the sum of all the integers in S.

Statement One Alone:

The range of S is a prime number that is less than 11 and is not a factor of 10.

Using information in statement one, we know that the range of S is 3 or 7. Thus, the average of S is also 3 or 7. However, since we don’t know whether it is 3 or 7, nor do we know the number of integers in S, statement one alone is not sufficient to answer the question.

Statement Two Alone:

S is composed of 5 different integers.

Since we don’t know any of the 5 integers, statement two alone is not sufficient to answer the question.

Statements One and Two Together:

From statement one, we know that the range and average are either both 3 or both 7. From statement two, we know S is composed of 5 different positive integers. Thus, the sum of these 5 integers is either 15 (if the average is 3) or 35 (if the average is 7). Therefore, we have two cases to consider: range = average = 3 (case 1) and range = average = 7 (case 2).

Case 1: range = average = 3

We can let x = the smallest number, so the largest number = x + 3. We can “squeeze” 2 more integers between x and x + 3, namely x + 1 and x + 2. So, there could be only 4 different total integers. However, remember that there should be 5 different integers in S; thus, case 1 is not possible.

So, it must be case 2: range = average = 7. If that is the case, the sum of the 5 integers is 35.

For example, the 5 integers could be 4, 5, 7, 8, and 11. We see that the range is 11 - 4 = 7 and the sum is 4 + 5 + 7 + 8 + 11 = 35 with an average of 7.

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