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# Set S consists of all the positive multiples of 5 that are less than K

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Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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18 Mar 2015, 05:12
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Set S consists of all the positive multiples of 5 that are less than K, and K is a positive integer not divisible by 5. The mean of Set S is not divisible by 5. Let N be the number of members of the set. N is not divisible by 5. What does N equal?

(1) N < 52
(2) K/5 > 48

Kudos for a correct solution.
[Reveal] Spoiler: OA

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Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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18 Mar 2015, 14:25
From the problem statement,

5n < k < 5n+5
n < k/5 < n+1
(k/5) - 1 < n <k/5

Since, mean of S is not divisible by 5, n has to be even.
and since n is not divisible by 5, therefore it is not a multiple of 10.

(1) n < 52
This is clearly INSUFFICIENT since n can be any even no. < 52 as long as it's not a multiple of 10

(2) k/5 > 48 => k/5-1 > 47 => n > 47
This is INSUFFICIENT as well since n could take any even value > 47 as long as it is not a multiple of 10.

From (1) , (2)

47 < n < 52

So the only possible value of n can be 48. Hence, both together are SUFFICIENT.

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Re: Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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23 Mar 2015, 06:24
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Bunuel wrote:
Set S consists of all the positive multiples of 5 that are less than K, and K is a positive integer not divisible by 5. The mean of Set S is not divisible by 5. Let N be the number of members of the set. N is not divisible by 5. What does N equal?

(1) N < 52
(2) K/5 > 48

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:

This is a tricky problem, because there are several constraints in the problem. first of all, the number of members is N, so the numbers in the set go from 5 to 5N. The integer K has the quality that 5N < K < (5N + 5), because K is bigger than the biggest member of the set, but 5N has to be the largest multiple of 5 less than K.

We know that the mean of the set is not divisible by 5, and this is important. For an evenly spaced set, the mean & median are identical. If there were an odd number of members, the mean & median would simply be the middle number on the list, and this of course would be some multiple of 5. The fact that mean & median is not divisible by 5 necessarily means that there must be an even number of members on the list. This way, the mean & median would be the average of the middle two numbers, which would be a non-integer, certainly not an integer divisible by 5. Therefore, N is an even number.

Finally, we are also told that N itself is not divisible by 5—an odd constraint that may be relevant.

Statement #1: Here, N could be any even number less than 52, as long as it’s not divisible by 5. It could be {48, 46, 44, 42, 38, ….} Multiple possibilities. This statement, alone and by itself, is not sufficient.

Statement #2: The statement tells us that K > 5*48, so it could be that N = 48, so that K would be between 5(48) and 5*(49). That’s a possibility, or N could be any larger even number that is not divisible by 5. N could be {48, 52, 54, 56, 68, 62, …} Again, multiple possibilities. This statement, alone and by itself, is not sufficient.

Combined statements: with both constraints, the only possible value if N = 48. The statements together are sufficient.

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Re: Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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15 Jan 2017, 22:24
I don't understand why n>47.
Why can't be N any even number less than 48? N could be also 40 or 32, why would it be wrong?
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Posts: 4
Re: Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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07 Aug 2017, 23:48
[quote="Bunuel"]Set S consists of all the positive multiples of 5 that are less than K, and K is a positive integer not divisible by 5. The mean of Set S is not divisible by 5. Let N be the number of members of the set. N is not divisible by 5. What does N equal?

(1) N < 52
(2) K/5 > 48

5N < K < (5N + 5)
I didnt understand how you got this equation
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Joined: 06 Jul 2016
Posts: 433
Location: Singapore
Concentration: Strategy, Finance
Re: Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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09 Aug 2017, 03:15
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machiavelli wrote:
Bunuel wrote:
Set S consists of all the positive multiples of 5 that are less than K, and K is a positive integer not divisible by 5. The mean of Set S is not divisible by 5. Let N be the number of members of the set. N is not divisible by 5. What does N equal?

(1) N < 52
(2) K/5 > 48

5N < K < (5N + 5)
I didnt understand how you got this equation

Set S has N terms in it, and K is the highest term, which is not a multiple of 5.

Set S = {5,10,15,.............,5N,K}
=> 5N is the highest multiple of 5 in the Set, and K is greater than 5N as per the constraints in the question
=> K > 5N
Since K has to be the largest term in the set, and 5N has to the largest multiple of 5 in the set, we can write the equation as
5N + 5 > K > 5N

E.g.
S = {5,10,15,20,21}
Here K = 21
5N = 20 (The largest multiple of 5 in the set) => N = 4 (# of Multiples of 5)

Does this help?
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Posts: 4
Re: Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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20 Feb 2018, 17:14
akshayk wrote:
machiavelli wrote:
Bunuel wrote:
Set S consists of all the positive multiples of 5 that are less than K, and K is a positive integer not divisible by 5. The mean of Set S is not divisible by 5. Let N be the number of members of the set. N is not divisible by 5. What does N equal?

(1) N < 52
(2) K/5 > 48

5N < K < (5N + 5)
I didnt understand how you got this equation

Set S has N terms in it, and K is the highest term, which is not a multiple of 5.

Set S = {5,10,15,.............,5N,K}
=> 5N is the highest multiple of 5 in the Set, and K is greater than 5N as per the constraints in the question
=> K > 5N
Since K has to be the largest term in the set, and 5N has to the largest multiple of 5 in the set, we can write the equation as
5N + 5 > K > 5N

E.g.
S = {5,10,15,20,21}
Here K = 21
5N = 20 (The largest multiple of 5 in the set) => N = 4 (# of Multiples of 5)

Does this help?

The question does not say that K is one of the terms of set S. It clearly says that S consist of all positive multiples of 5, which are less than K. In your example above, S will be S={5,10,15,20} and K will be 21, separate from the set.
I get the point that K/5>48, so K>5x48, so there should be at least 49 terms in K, greater than 48. But I don't understand why official answer says K could equal to at least 48, there is NO >= sign!
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Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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20 Feb 2018, 22:48
Set S has N terms in it, and K is the highest term, which is not a multiple of 5.

Set S = {5,10,15,.............,5N,K}
=> 5N is the highest multiple of 5 in the Set, and K is greater than 5N as per the constraints in the question
=> K > 5N
Since K has to be the largest term in the set, and 5N has to the largest multiple of 5 in the set, we can write the equation as
5N + 5 > K > 5N

E.g.
S = {5,10,15,20,21}
Here K = 21
5N = 20 (The largest multiple of 5 in the set) => N = 4 (# of Multiples of 5)

Does this help?[/quote]

The question does not say that K is one of the terms of set S. It clearly says that S consist of all positive multiples of 5, which are less than K. In your example above, S will be S={5,10,15,20} and K will be 21, separate from the set.
I get the point that K/5>48, so K>5x48, so there should be at least 49 terms in K, greater than 48. But I don't understand why official answer says K could equal to at least 48, there is NO >= sign![/quote]

Hi

The official answer does not say that K is equal to at least 48, it rather says that 'N' is equal to at least 48.
If K > 5*48, it means the positive multiples of 5 in K are at least 48 (starting from 5*1, 5*2, 5*3,..... to ... 5*48 at least).

And i agree with your first point, the example which was taken b{5, 10, 15, 20, 21} should not involve 21.. It should be {5, 10, 15, 20} and K=21 has to be separate from this set.
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Re: Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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20 Feb 2018, 23:01
1
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nevermind. I finally understood this! Let me try to explain it-

Set S consists of all the positive multiples of 5 that are less than K, and K is a positive integer not divisible by 5. The mean of Set S is not divisible by 5. Let N be the number of members of the set. N is not divisible by 5. What does N equal?

(1) N < 52
(2) K/5 > 48

so set S could look like {5,10,15,20,25,.....5*N} where N is the total number of elements in the set S.
We can also write this set as {5*1, 5*2, 5*3, 5*4, 5*5,.....5*N}. We have given that the last number which is 5*N is less than K. So we can write that K>5*N. So for example, if our set consists of only 4 numbers, i.e, {5,10,15,20} the K could be any number greater than 20 and not a multiple of 5 (or divisible by 5).
Another thing to notice here is that mean of set S is not divisible by 5. Since its a set of all numbers that are at a constant distance of 5, the mean could be the middle number if a total number of elements are odd. For example, if the set has 3 terms, {5,10,15} the mean could be middle number 10. So if the set S is not divisible by 5, it means it must have even number of terms. So N is even.

So far we got-
N is even
K>5*N
We have to find the value of N?

Statement 1)
N<52, so N could be any even number less than 52. NOT SUFFICIENT

Statement 2) K/5 > 48
or we can write: K > 5*48
Since we know that K>5*N and from above we get K>5*48, this suggests that N could 48 or any even number greater than 48, not divisible by 5. So N could be 48, 52, 54 etc.
NOT SUFFICIENT

Taking 1) and 2) together,

From one we get N<52 and from statement 2, we get N could be 48 or higher
so possible values of N could be 48, 49, 50, 51, but since N has to be even and not divisible by 5, the only value we are left with is 48. Thus answer (C).
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Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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20 Feb 2018, 23:03
amanvermagmat wrote:
Set S has N terms in it, and K is the highest term, which is not a multiple of 5.

Set S = {5,10,15,.............,5N,K}
=> 5N is the highest multiple of 5 in the Set, and K is greater than 5N as per the constraints in the question
=> K > 5N
Since K has to be the largest term in the set, and 5N has to the largest multiple of 5 in the set, we can write the equation as
5N + 5 > K > 5N

E.g.
S = {5,10,15,20,21}
Here K = 21
5N = 20 (The largest multiple of 5 in the set) => N = 4 (# of Multiples of 5)

Does this help?

The question does not say that K is one of the terms of set S. It clearly says that S consist of all positive multiples of 5, which are less than K. In your example above, S will be S={5,10,15,20} and K will be 21, separate from the set.
I get the point that K/5>48, so K>5x48, so there should be at least 49 terms in K, greater than 48. But I don't understand why official answer says K could equal to at least 48, there is NO >= sign![/quote]

Hi

The official answer does not say that K is equal to at least 48, it rather says that 'N' is equal to at least 48.
If K > 5*48, it means the positive multiples of 5 in K are at least 48 (starting from 5*1, 5*2, 5*3,..... to ... 5*48 at least).

And i agree with your first point, the example which was taken b{5, 10, 15, 20, 21} should not involve 21.. It should be {5, 10, 15, 20} and K=21 has to be separate from this set.[/quote]

Hi amanvermagmat

Let $$S = {5,10,......., 5n}$$
$$5n < K$$
As per question, Mean is not divisible by 5. May I know how is it possible?

Sum of the Set S = $$\frac{n(5+5n)}{2} = \frac{5n(n+1)}{2}$$
Mean = $$\frac{5(n+1)}{2}$$... So irrespective of n, mean is always divisible by 5.

Am I Missing anything here?
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Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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20 Feb 2018, 23:28
rahul16singh28 wrote:
amanvermagmat wrote:
Set S has N terms in it, and K is the highest term, which is not a multiple of 5.

Set S = {5,10,15,.............,5N,K}
=> 5N is the highest multiple of 5 in the Set, and K is greater than 5N as per the constraints in the question
=> K > 5N
Since K has to be the largest term in the set, and 5N has to the largest multiple of 5 in the set, we can write the equation as
5N + 5 > K > 5N

E.g.
S = {5,10,15,20,21}
Here K = 21
5N = 20 (The largest multiple of 5 in the set) => N = 4 (# of Multiples of 5)

Does this help?

The question does not say that K is one of the terms of set S. It clearly says that S consist of all positive multiples of 5, which are less than K. In your example above, S will be S={5,10,15,20} and K will be 21, separate from the set.
I get the point that K/5>48, so K>5x48, so there should be at least 49 terms in K, greater than 48. But I don't understand why official answer says K could equal to at least 48, there is NO >= sign!

Hi

The official answer does not say that K is equal to at least 48, it rather says that 'N' is equal to at least 48.
If K > 5*48, it means the positive multiples of 5 in K are at least 48 (starting from 5*1, 5*2, 5*3,..... to ... 5*48 at least).

And i agree with your first point, the example which was taken b{5, 10, 15, 20, 21} should not involve 21.. It should be {5, 10, 15, 20} and K=21 has to be separate from this set.[/quote]

Hi amanvermagmat

Let $$S = {5,10,......., 5n}$$
$$5n < K$$
As per question, Mean is not divisible by 5. May I know how is it possible?

Sum of the Set S = $$\frac{n(5+5n)}{2} = \frac{5n(n+1)}{2}$$
Mean = $$\frac{5(n+1)}{2}$$... So irrespective of n, mean is always divisible by 5.

Am I Missing anything here?[/quote]

Hi Rahul

Yes you are correct that mean is = 5*(n+1)/2
But this is NOT always divisible by 5. Because if n is odd, then (n+1) is even and divisible by 2. The mean now is divisible by 5 as you said.
BUT if n is even, then (n+1) is odd, and (n+1) will NOT be divisible by 2, thus (n+1)/2 will give a decimal, 5*(n+1)/2 will thus give a decimal value, and thus NOT divisible by 5.

You can check with any example where you take n as even. Eg, if n=4, then {5, 10, 15, 20}. Mean is 12.5, NOT divisible by 5
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Posts: 53
Re: Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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02 Mar 2018, 03:30
2nd condition says k>240 i.e. ,k can be 241,242.....
1st says n<52
@k=241 you get 48 multiples of 5, hence you can have 48,49,50 or 51 as ans ( combining the two equations)

Mean = 5 (n+1)/2
At n= 49 and 51, n+1 gets even and mean gets divisible by 2.
Hence either n is 48 or 50.
N can not be 50 as at n=50 n gets divisible by 5 which can not happen as per question.

Hence n=48 .

Ans C
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Posts: 2
Re: Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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13 Mar 2018, 05:25
From "Set S consists of all positive multiples of 5..."Do we just assume that the set is evenly spaced unless stated otherwise? Was thinking of other possibilities too like {5,5,5,..}
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Posts: 900
Location: India
Re: Set S consists of all the positive multiples of 5 that are less than K [#permalink]

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13 Mar 2018, 10:54
isabelleo wrote:
From "Set S consists of all positive multiples of 5..."Do we just assume that the set is evenly spaced unless stated otherwise? Was thinking of other possibilities too like {5,5,5,..}

Hello

Yes, a set having positive multiples of 5 means an evenly spaced set only. And the question clearly states, "..consists of all the positive multiples of 5 that are less than K". So for any given positive value of k, we will have to consider ALL positive multiples of 5 before that. That obviously cannot mean something like {5,5,5,..}

Eg, if k=27, then S = {5, 10, 15, 20, 25}
Re: Set S consists of all the positive multiples of 5 that are less than K   [#permalink] 13 Mar 2018, 10:54
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# Set S consists of all the positive multiples of 5 that are less than K

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