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# Are all of the numbers in a certain list of 15 numbers equal?

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Intern
Joined: 16 Mar 2011
Posts: 9

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Re: Number properties from OG 12 DS133 [#permalink]

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16 Apr 2011, 03:57
00:00

Difficulty:

35% (medium)

Question Stats:

83% (00:35) correct 17% (02:05) wrong based on 12 sessions

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Quote:
Are all of the numbers in a certain list of 15 numbers equal?

(1) The sum of all the numbers in the list is 60.

(2) The sum of any 3 numbers in the list is 12.

My explanation is similar to Entwistle's explanation.

Here's my explanation.

(1)- Trivial Case: All numbers in the list are 4.
Alternative Case: All but two numbers in the list are 4; the others are 3 and 5. Please that this alternative case is only one of many alternative cases; for example, all but 3 numbers could be 4, and the remaining three numbers could be 3,4, and 5.

INSUFFICIENT
(2) implies that for numbers a,b,c,d,e,f,g,h,i,j,k,l,m,n,o, in the the list of 15 numbers,
a+b+c=a+b+d--> c=d
a+b+d=a+b+e--> d=e
a+b+e=a+b+f--> e=f
...a+b+n=a+b+o--> o=n=m=l=k=j=i=h=g=f=e=d=c
To cover a and b, note: c+d+e=b+d+e-->b=c
and a+d+e=b+d+e-->a=b
Thus all numbers in the list are equal.

SUFFICIENT.

B
[Reveal] Spoiler: OA

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Math Expert
Joined: 02 Sep 2009
Posts: 41892

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Re: Are all of the numbers in a certain list of 15 numbers [#permalink]

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14 Aug 2017, 02:22
Are all of the numbers in a certain list of 15 numbers equal?

(1) The sum of all the numbers in the list is 60. Clearly insufficient.

(2) The sum of any 3 numbers in the list is 12. Since the sum of ANY 3 numbers is 12 then ALL numbers must equal to 12/3=4, because if not all the numbers equal to 4, then we could pick certain set of 3 numbers so that their sum is not 12. Sufficient.

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Kudos [?]: 129160 [0], given: 12194

Re: Are all of the numbers in a certain list of 15 numbers   [#permalink] 14 Aug 2017, 02:22
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