P = {6, 3, 0, d, 4, 14, 9, 2d} If d is the smallest positive integer

Question

P = {6, 3, 0, d, 4, 14, 9, 2d}

If d is the smallest positive integer such that the range of the remainders obtained when multiples of 3 are divided by d is 3, by what percentage is the median of the numbers in P smaller than the mean of the numbers in P?

A. 11.1%
B. 12.5%
C. 16.7%
D. 20.0%
E. Cannot Be Determined

Palladin · Answer

If the range of remainders is  3 , the minimum possible value for  d  is  4 .

So,  P = \{6, 3, 0, 4, 4, 14, 9, 8\} 
In ascending order,  P = \{0, 3, 4, 4, 6, 8, 9, 14\}

Median(P) =  5 
Mean (P) =  6

Percentage by which median is less than mean =  \frac{6-5}{6}*100  =  16.67\%

Answer is  C .

Luca1111111111111 · Answer

Could you please explain how to get there: "If the range of remainders is  3 , the minimum possible value for  d  is  4 ."? 
Thank you

firas92 · Answer

Any division by 4 can give a remainder 0,1, 2 or 3. So the range of remainders of division by 4 is 3.

Now lets check if any multiple of 3 leaves a reminder 3 when divided by 4 so that we can confirm that d must be 4.

We have 15 which leaves a remainder 3 when divided by 4

So yes d must be 4.

On the other hand, no positive integer less than 4 can have the range of remainders as 3 so you know that you have to start testing from 4

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HailieBui · Answer

if the range of the remainders is 3 then we can have either (0,3)(1,4)(2,5)... and so on. However, the higher the max value of remainder, the higher the divisor (since remainder must be smaller than divisor) so we would have to choose the smallest pair (0,3) and the smallest integer bigger than 3 is 4.
P={6,3,0,4,4,14,9,8}
P reordered= {0,3,4,4,6,8,9,18}
Median= (6+4)/2=5
Mean= sum(0,3,4,4,6,8,9,18)/8=6
(6-5)/6=1/6=16.7%

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rheam25 · Answer

MathRevolution  could you please help solve this?

Kritisood · Answer

@experts how can d be 4?
3/4 R=3
6/4 R=2 
9/4 R=1
Range = 3-1=2

Bunuel   chetan2u   GMATinsight  could you please assist?

GMATinsight · Answer

Kritisood Luca1111111111111 CONCEPT : If consecutive numbers are divided by n then remainders can be values from 0 to (n-1) e.g. When the integers {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ---) are divided by n=2, remainders are {0, 1} n=3, remainders are {0, 1, 2} n=4, remainders are {0, 1, 2, 3} n=5, remainders are {0, 1, 2, 3, 5} and so on... Now, Range of Remainders for divisor d = Highest - Lowest = (d-1) - 0 = 3 i.e. d = 4 CROSS CHECK Remainder when 3 is divided by 4 = 3 Remainder when 6 is divided by 4 = 2 Remainder when 9 is divided by 4 = 1 Remainder when 12 is divided by 4 = 0 Remainder when 15 is divided by 4 = 3 i.e. now the remainders are cyclic and range of remainders = 3-0 = 3 I hope this help!

Kritisood · Answer

GMATinsight  thanks for the response, so when the question says "d is the smallest positive integer such that the range of the remainders obtained  when multiples of 3 "  it is talking about   any   multiples of three and not the ones part of the P set "P = {6, 3, 0, d, 4, 14, 9, 2d}"?

GMATinsight · Answer

Hi Kritisood Yes you are right about this statement. The statement has been made in isolation so teh two pieces of information need to be processed independently.

kittle · Answer

BUT HOW IS RANGE 3 of this set? - P = {6, 3, 0, d, 4, 14, 9, 2d}

amoghhlgr · Answer

I guess we are struggling to understand that 0 is also a multiple of 3, and 0 is a part of the set. So, that 0 divided by 4 is 0... Hence, the range will be 3-0... the difference between highest remainder from dividing 3 by 4 and the lowest remainder from dividing 0 by 4.

For clarification, all the multiples of 3 given in the set are {0,3,6,9}

Hope that helps!

kittle · Answer

Hey- I think I decoded this question
Basically, this is a takeaway going forward: If we know the range of remainders for a particular divisor D, then D= R+1, where R is the range of its remainders
So let us say the range R for a divisor D, is 5-> then we can dedude that the number would be 6

Bunuel  - do you agree as well on my deduction

Best, Kittle!

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P = {6, 3, 0, d, 4, 14, 9, 2d} If d is the smallest positive integer

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P = {6, 3, 0, d, 4, 14, 9, 2d} If d is the smallest positive integer

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