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n is the product of least and greatest 6 consecutive integers. What is

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anuragsingal
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n is the product of least and greatest 6 consecutive integers. What is [#permalink] New post   Updated on: 05 Jan 2015, 04:15
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n is the product of least and greatest 6 consecutive integers. What is n?

(1) The greatest integer is 20
(2) The average arithmetic mean of 6 consecutive integers is 17.5

Solution:
Show Spoiler
If n is the product of the least and the greatest of 6 consecutive integers, what is the value of n?
What do we know? We have a list of 6 consecutive integers. So, if we can determine which integers make up our list, we can certainly answer the question.

(1) the greatest integer in the list is 20
Well, if we know the biggest number on the list we can certainly count backwards to determine the other 5: sufficient.
(2) the average (arithmetic mean) of the integers is 17.5
Since our numbers are consecutive, we can certainly use this information to figure out exactly what the list is: sufficient.

If we needed to actually do so, we could:
1) know that for a set of consecutive numbers, mean = median. Since we have an even number of terms, the median is the average of the two middle terms, so the two middle terms in our set must be 17 and 18, which we can then expand to {15, 16, 17, 18, 19, 20}; or
2) use the average formula.
Average = (sum of terms)/(# of terms)
17.5 = (t1 + t2 + t3 + t4 + t5 + t6)/6
105 = t1 + t2 + t3 + t4 + t5 + t6
And, since our terms are consecutive, we know that:
t1 + t2 + t3 + t4 + t5 + t6 = t1 + (t1 + 1) + (t1 + 2) + (t1 + 3) + (t1 + 4) + (t1 + 5)
so
105 = 6(t1) + 15
90 = 6(t1)
15 = t1
so our set must be {15, 16, 17, 18, 19, 20}
Lots of other tricks we could use to also figure out the exact set.
Each of (1) and (2) is sufficient alone: choose D.
ShowHide Answer
Official Answer
D

Originally posted by anuragsingal on 11 Sep 2010, 04:53.
Last edited by Bunuel on 05 Jan 2015, 04:15, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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anuragsingal
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Re: n is the product of least and greatest 6 consecutive integers. What is [#permalink] New post   11 Sep 2010, 04:57
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106. n is the product of least and greatest 6 consecutive integers. What is n?
1) the greatest integer is 20
2) the average arithmetic mean of 6 consecutive integers is 17.5

Solution:
If n is the product of the least and the greatest of 6 consecutive integers, what is the value of n?
What do we know? We have a list of 6 consecutive integers. So, if we can determine which integers make up our list, we can certainly answer the question.

(1) the greatest integer in the list is 20
Well, if we know the biggest number on the list we can certainly count backwards to determine the other 5: sufficient.
(2) the average (arithmetic mean) of the integers is 17.5
Since our numbers are consecutive, we can certainly use this information to figure out exactly what the list is: sufficient.

If we needed to actually do so, we could:
1) know that for a set of consecutive numbers, mean = median. Since we have an even number of terms, the median is the average of the two middle terms, so the two middle terms in our set must be 17 and 18, which we can then expand to {15, 16, 17, 18, 19, 20}; or
2) use the average formula.
Average = (sum of terms)/(# of terms)
17.5 = (t1 + t2 + t3 + t4 + t5 + t6)/6
105 = t1 + t2 + t3 + t4 + t5 + t6
And, since our terms are consecutive, we know that:
t1 + t2 + t3 + t4 + t5 + t6 = t1 + (t1 + 1) + (t1 + 2) + (t1 + 3) + (t1 + 4) + (t1 + 5)
so
105 = 6(t1) + 15
90 = 6(t1)
15 = t1
so our set must be {15, 16, 17, 18, 19, 20}
Lots of other tricks we could use to also figure out the exact set.
Each of (1) and (2) is sufficient alone: choose D.
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ramgmat
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Re: n is the product of least and greatest 6 consecutive integers. What is [#permalink] New post   11 Sep 2010, 04:55
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Hi anurag,

Could you please change the colour of the font. For some reason the top part shows yellow....Maybe its only in my system.

Thanks. :)
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Re: n is the product of least and greatest 6 consecutive integers. What is [#permalink] New post   24 Sep 2010, 03:18
1
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anuragsingal wrote:
106. n is the product of least and greatest 6 consecutive integers. What is n?
1) the greatest integer is 20
2) the average arithmetic mean of 6 consecutive integers is 17.5

Solution:
If n is the product of the least and the greatest of 6 consecutive integers, what is the value of n?
What do we know? We have a list of 6 consecutive integers. So, if we can determine which integers make up our list, we can certainly answer the question.

(1) the greatest integer in the list is 20
Well, if we know the biggest number on the list we can certainly count backwards to determine the other 5: sufficient.
(2) the average (arithmetic mean) of the integers is 17.5
Since our numbers are consecutive, we can certainly use this information to figure out exactly what the list is: sufficient.

If we needed to actually do so, we could:
1) know that for a set of consecutive numbers, mean = median. Since we have an even number of terms, the median is the average of the two middle terms, so the two middle terms in our set must be 17 and 18, which we can then expand to {15, 16, 17, 18, 19, 20}; or
2) use the average formula.
Average = (sum of terms)/(# of terms)
17.5 = (t1 + t2 + t3 + t4 + t5 + t6)/6
105 = t1 + t2 + t3 + t4 + t5 + t6
And, since our terms are consecutive, we know that:
t1 + t2 + t3 + t4 + t5 + t6 = t1 + (t1 + 1) + (t1 + 2) + (t1 + 3) + (t1 + 4) + (t1 + 5)
so
105 = 6(t1) + 15
90 = 6(t1)
15 = t1
so our set must be {15, 16, 17, 18, 19, 20}
Lots of other tricks we could use to also figure out the exact set.
Each of (1) and (2) is sufficient alone: choose D.



Anurag:

You do not have to sum up all the 6 #s (t1,t2....) to find iut the #s for a given average.
As it is given that the 6#s are cosecutive intergers, the average of those 6 #s shud be th average of the middle 2 numbers ==> sum of middle 2 #s = 35 i.e 17.5*2 ==> middle #s, which are consecutive are 17 and 18 hence the first 2 are 15 and 16 and last two #s are 19 and 20.

Hope it helps
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Re: n is the product of least and greatest 6 consecutive integers. What is [#permalink] New post   11 Sep 2010, 21:28
@Moderators -- Please move this post to the DS forum.
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Re: n is the product of least and greatest 6 consecutive integers. What is [#permalink] New post   24 Sep 2010, 01:18
anuragsingal wrote:
106. n is the product of least and greatest 6 consecutive integers. What is n?
1) the greatest integer is 20
2) the average arithmetic mean of 6 consecutive integers is 17.5

Solution:
If n is the product of the least and the greatest of 6 consecutive integers, what is the value of n?
What do we know? We have a list of 6 consecutive integers. So, if we can determine which integers make up our list, we can certainly answer the question.

(1) the greatest integer in the list is 20
Well, if we know the biggest number on the list we can certainly count backwards to determine the other 5: sufficient.
(2) the average (arithmetic mean) of the integers is 17.5
Since our numbers are consecutive, we can certainly use this information to figure out exactly what the list is: sufficient.

If we needed to actually do so, we could:
1) know that for a set of consecutive numbers, mean = median. Since we have an even number of terms, the median is the average of the two middle terms, so the two middle terms in our set must be 17 and 18, which we can then expand to {15, 16, 17, 18, 19, 20}; or
2) use the average formula.
Average = (sum of terms)/(# of terms)
17.5 = (t1 + t2 + t3 + t4 + t5 + t6)/6
105 = t1 + t2 + t3 + t4 + t5 + t6
And, since our terms are consecutive, we know that:
t1 + t2 + t3 + t4 + t5 + t6 = t1 + (t1 + 1) + (t1 + 2) + (t1 + 3) + (t1 + 4) + (t1 + 5)
so
105 = 6(t1) + 15
90 = 6(t1)
15 = t1
so our set must be {15, 16, 17, 18, 19, 20}
Lots of other tricks we could use to also figure out the exact set.
Each of (1) and (2) is sufficient alone: choose D.


First of all, do post the questions in the relevant forums.
Secondly the question is badly written it should say :

n is the product of least and greatest integers of the 6 consecutive integers. What is n?
1) the greatest integer is 20
2) the average arithmetic mean of 6 consecutive integers is 17.5
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Re: n is the product of least and greatest 6 consecutive integers. What is [#permalink] New post   31 Jan 2016, 22:11
Expert Reply
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.

n is the product of least and greatest 6 consecutive integers. What is n?

(1) The greatest integer is 20
(2) The average arithmetic mean of 6 consecutive integers is 17.5


In the original condition, you need to figure out the number of consecutive integers and the first integer. So, there are 2 variables. In this question, it’s given that there are 6 consecutive integers and you only need to figure out the first integer. Thus, there is 1 variable, which should match with the number of equations. So you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.
For 1), 15,16,17,18,19,20, which is unique and sufficient.
For 2), also 15,16,17,18,19,20, which is unique and sufficient.
Therefore, the answer is D.


 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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Re: n is the product of least and greatest 6 consecutive integers. What is [#permalink] New post   21 Dec 2016, 21:59
anuragsingal wrote:
106. n is the product of least and greatest 6 consecutive integers. What is n?
1) the greatest integer is 20
2) the average arithmetic mean of 6 consecutive integers is 17.5

Solution:
If n is the product of the least and the greatest of 6 consecutive integers, what is the value of n?
What do we know? We have a list of 6 consecutive integers. So, if we can determine which integers make up our list, we can certainly answer the question.

(1) the greatest integer in the list is 20
Well, if we know the biggest number on the list we can certainly count backwards to determine the other 5: sufficient.
(2) the average (arithmetic mean) of the integers is 17.5
Since our numbers are consecutive, we can certainly use this information to figure out exactly what the list is: sufficient.

If we needed to actually do so, we could:
1) know that for a set of consecutive numbers, mean = median. Since we have an even number of terms, the median is the average of the two middle terms, so the two middle terms in our set must be 17 and 18, which we can then expand to {15, 16, 17, 18, 19, 20}; or
2) use the average formula.
Average = (sum of terms)/(# of terms)
17.5 = (t1 + t2 + t3 + t4 + t5 + t6)/6
105 = t1 + t2 + t3 + t4 + t5 + t6
And, since our terms are consecutive, we know that:
t1 + t2 + t3 + t4 + t5 + t6 = t1 + (t1 + 1) + (t1 + 2) + (t1 + 3) + (t1 + 4) + (t1 + 5)
so
105 = 6(t1) + 15
90 = 6(t1)
15 = t1
so our set must be {15, 16, 17, 18, 19, 20}
Lots of other tricks we could use to also figure out the exact set.
Each of (1) and (2) is sufficient alone: choose D.



When they say 'Consecutive numbers' do we assume that they are normal consecutive numbers (2,3,4,5,etc) rather than even / odd consecutive numbers?
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Re: n is the product of least and greatest 6 consecutive integers. What is [#permalink] New post   22 Dec 2016, 01:58
Expert Reply
nischaynshah wrote:
anuragsingal wrote:
106. n is the product of least and greatest 6 consecutive integers. What is n?
1) the greatest integer is 20
2) the average arithmetic mean of 6 consecutive integers is 17.5

Solution:
If n is the product of the least and the greatest of 6 consecutive integers, what is the value of n?
What do we know? We have a list of 6 consecutive integers. So, if we can determine which integers make up our list, we can certainly answer the question.

(1) the greatest integer in the list is 20
Well, if we know the biggest number on the list we can certainly count backwards to determine the other 5: sufficient.
(2) the average (arithmetic mean) of the integers is 17.5
Since our numbers are consecutive, we can certainly use this information to figure out exactly what the list is: sufficient.

If we needed to actually do so, we could:
1) know that for a set of consecutive numbers, mean = median. Since we have an even number of terms, the median is the average of the two middle terms, so the two middle terms in our set must be 17 and 18, which we can then expand to {15, 16, 17, 18, 19, 20}; or
2) use the average formula.
Average = (sum of terms)/(# of terms)
17.5 = (t1 + t2 + t3 + t4 + t5 + t6)/6
105 = t1 + t2 + t3 + t4 + t5 + t6
And, since our terms are consecutive, we know that:
t1 + t2 + t3 + t4 + t5 + t6 = t1 + (t1 + 1) + (t1 + 2) + (t1 + 3) + (t1 + 4) + (t1 + 5)
so
105 = 6(t1) + 15
90 = 6(t1)
15 = t1
so our set must be {15, 16, 17, 18, 19, 20}
Lots of other tricks we could use to also figure out the exact set.
Each of (1) and (2) is sufficient alone: choose D.



When they say 'Consecutive numbers' do we assume that they are normal consecutive numbers (2,3,4,5,etc) rather than even / odd consecutive numbers?


"Consecutive integers" ALWAYS mean integers that follow each other in order with common difference of 1: ... x-3, x-2, x-1, x, x+1, x+2, ....

For example:

-7, -6, -5 are consecutive integers.

2, 4, 6 ARE NOT consecutive integers, they are consecutive even integers.

3, 5, 7 ARE NOT consecutive integers, they are consecutive odd integers.
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Re: n is the product of least and greatest 6 consecutive integers. What is [#permalink] New post   22 Dec 2016, 02:30
Did someone actually come across cosecutive even/odd integers in the GMAT? I know the GMAT OG introduces the theory shortly but I never saw a question where we think about them. Only (normal) consecutive integers ?!?!?!?
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n is the product of least and greatest 6 consecutive integers. What is [#permalink] New post   10 Jan 2017, 23:16
Expert Reply
nischaynshah wrote:
anuragsingal wrote:
106. n is the product of least and greatest 6 consecutive integers. What is n?
1) the greatest integer is 20
2) the average arithmetic mean of 6 consecutive integers is 17.5

Solution:
If n is the product of the least and the greatest of 6 consecutive integers, what is the value of n?
What do we know? We have a list of 6 consecutive integers. So, if we can determine which integers make up our list, we can certainly answer the question.

(1) the greatest integer in the list is 20
Well, if we know the biggest number on the list we can certainly count backwards to determine the other 5: sufficient.
(2) the average (arithmetic mean) of the integers is 17.5
Since our numbers are consecutive, we can certainly use this information to figure out exactly what the list is: sufficient.

If we needed to actually do so, we could:
1) know that for a set of consecutive numbers, mean = median. Since we have an even number of terms, the median is the average of the two middle terms, so the two middle terms in our set must be 17 and 18, which we can then expand to {15, 16, 17, 18, 19, 20}; or
2) use the average formula.
Average = (sum of terms)/(# of terms)
17.5 = (t1 + t2 + t3 + t4 + t5 + t6)/6
105 = t1 + t2 + t3 + t4 + t5 + t6
And, since our terms are consecutive, we know that:
t1 + t2 + t3 + t4 + t5 + t6 = t1 + (t1 + 1) + (t1 + 2) + (t1 + 3) + (t1 + 4) + (t1 + 5)
so
105 = 6(t1) + 15
90 = 6(t1)
15 = t1
so our set must be {15, 16, 17, 18, 19, 20}
Lots of other tricks we could use to also figure out the exact set.
Each of (1) and (2) is sufficient alone: choose D.



When they say 'Consecutive numbers' do we assume that they are normal consecutive numbers (2,3,4,5,etc) rather than even / odd consecutive numbers?



Hi,

In the original condition, from n,n+1,n+2,n+3,n+4,n+5, there is 1 variable(n).
Hence, D is likely to be an answer.

Happy Studying!
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Re: n is the product of least and greatest 6 consecutive integers. What is [#permalink] New post   19 Oct 2020, 19:40
n is the product of least and greatest 6 consecutive integers. What is n?

For the sake of my own learning..

if n is the product of 'the' least and greatest of 6 consecutive integers,

\(n = (x)*(x+5)\) where \(x\) is the smallest number and \(x+5\) is the largest.

(1) The greatest integer is 20

\(20 = x+5\). Solve for x, then solve for n above. sufficient

(2) The average arithmetic mean of 6 consecutive integers is 17.5

\(17.5= \frac{5x + 15 }{ 6} \)

Solve for x, then solve for n above. sufficient
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Re: n is the product of least and greatest 6 consecutive integers. What is [#permalink]
19 Oct 2020, 19:40
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