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# If a+b+c+d=30, where a,b,c and d are integers, then the minimum possib

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Director
Joined: 19 Oct 2018
Posts: 880
Location: India
If a+b+c+d=30, where a,b,c and d are integers, then the minimum possib  [#permalink]

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30 Apr 2019, 09:53
2
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Difficulty:

45% (medium)

Question Stats:

58% (02:05) correct 42% (02:54) wrong based on 26 sessions

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If a+b+c+d=30, where a,b,c and d are integers, then the minimum possible value of $$(a - b)^2$$ + $$(a - c)^2$$ + $$(a - d)^2$$ is

A. 3
B. 2
C. 0
D. 1
E. 5
Manager
Status: Don't Give Up!
Joined: 15 Aug 2014
Posts: 95
Location: India
Concentration: Operations, General Management
GMAT Date: 04-25-2015
WE: Engineering (Manufacturing)
Re: If a+b+c+d=30, where a,b,c and d are integers, then the minimum possib  [#permalink]

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30 Apr 2019, 10:04
Superb question. Simple but tricky.

To minimize the overall value individual values of a-b , a-c, a-d should be minimum and that is possible only when they are same digit or minimum difference.
Values would be 7 8 8 7 which gives answer 2

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Intern
Joined: 27 Nov 2018
Posts: 36
GMAT 1: 510 Q33 V28
Re: If a+b+c+d=30, where a,b,c and d are integers, then the minimum possib  [#permalink]

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30 Apr 2019, 10:50
nick1816 wrote:
If a+b+c+d=30, where a,b,c and d are integers, then the minimum possible value of $$(a - b)^2$$ + $$(a - c)^2$$ + $$(a - d)^2$$ is

A. 3
B. 2
C. 0
D. 1
E. 5

IMO - B

$$(a - b)^2$$ + $$(a - c)^2$$ + $$(a - d)^2$$ - Since in the question the difference of digits is squared, so the min value can be zero. It cant be negative.
Lets assume the min value is zero. It means a=b=c=d
=> 4a = 30
=> a= 30/4 = 15/2 = 7.5 But 'a' cant be a decimal number as the question states that the digits are integers. So, lets assume that a=7 or a=8
In case, a= 7, sum of all digits is 7 x 4 = 28 ( Assuming all digits to be of same value so as to get zero as the difference).
It falls short of 30 by 02. So, we need to increase two 07 into two eights. Hence the numbers are 7,7,8,8

In case, a=8, sum of all digits is 8 x 4 = 32 ( Assuming all digits to be of same value so as to get zero as the difference).
It exceeds 30 by 02. So, we need to decrease two 08 into two seven. Hence the numbers are 8 , 8 , 7 , 7

Now, (8-8)^2 + (8-7)^2 + (8-7)^2 = 0 + 1 + 1 = 2 Answer
Re: If a+b+c+d=30, where a,b,c and d are integers, then the minimum possib   [#permalink] 30 Apr 2019, 10:50
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