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

Question

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

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

Bunuel · Accepted Answer

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.

Answer = (C)

rdoshi · Answer

From the problem statement, 5n < k < 5n+5 n < k/5 < n+1 (k/5) - 1 < n 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. Answer is C.

MarcoL90 · Answer

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?

machiavelli · Answer

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

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

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

akshayk · Answer

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?

gmatninja07 · Answer

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!

amanvermagmat · Answer

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.

gmatninja07 · Answer

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).

rahul16singh28 · Answer

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.

Hi  amanvermagmat

Can you please help me to clear my doubt.

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?

amanvermagmat · Answer

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.

Hi  amanvermagmat

Can you please help me to clear my doubt.

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?

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

Timebomb · Answer

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

isabelleo · Answer

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,..}

amanvermagmat · Answer

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}

Jane121393 · Answer

gmatninja07 I can't understand why you can be sure that K > 5*48 and K > 5*N => N must be 48 or greater. I think N could be any value. I do not see any relationship between 5*48 and 5*N. For example, N could be 47, then 5*47 < 5*48 < K. Could you pls explain this point. Thanks so much.

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

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