GMATGuruNY wrote:Mathsbuddy wrote:To simplify what I wrote in the last posting:
(A+B+C) is a factor of the sum of the integers in list L
and
(A+B+C) is a positive integer
Therefore (A+B+C) has to be included in "all the positive integers that MUST be factors of the sum of the integers in list L?"
Ditto for any other factors of (A+B+C)
As shown in my initial post, S = 111(A+B+C).
List L is composed of every positive integer that MUST be a factor of S.
Must means IN EVERY CASE.
Case 1: A=1, B=2, C=3
Here, S = 111(1+2+3) = 111*7.
Factors of 111*7 are
1, 3, 7,
37, 111.
Case 2: A=1, B=2, C=8
Here, S = 111(1+2+8) = 111*11.
Factors of 111*11 are
1, 3, 11,
37, 111.
Case 3: A=1, B=4, C=8
Here, Here, S = 111(1+4+8) = 111*13.
Factors of 111*13 are
1, 3, 13,
37, 111.
As the cases above illustrate, only the values in red MUST be factors of S.
Since 7 is NOT a factor in Cases 2 and 3, it is not true that 7 MUST be a factor of S.
Since 11 is NOT a factor in Cases 1 and 3, it is not true that 11 MUST be a factor of S.
Since 13 is NOT a factor in Cases 1 and 2, it is not true that 13 MUST be a factor of S.
Result:
L = {1, 3, 37, 111}.
Sum = 1+3+37+111 = 152.
Thank you very much for this. I have slept on it and now see how "MUST" indicates this restriction; and I understand fully what you mean.
However, I do believe that the "MUST" can still be interpretted 2 different ways.
If the question had begun:
"L is
the list of...", then I would agree 100%.
"L is
a list of...", then I would not agree, as MUST could apply to that one singular (undetermined) case.
As the question stands, it is not clear whether L covers all cases, or L is just one particular (unspecified) case.
For example, consider the question:
List L: 123, 231, 312
In list L above, there are 3 positive integers, where each digit is different and nonzero. Which of the following is the sum of all the positive integers that MUST be factors of the sum of the integers in list L?
This does not conflict with the given question, therefore the possibility of a individual case cannot be ignored - even with "must" present. Here the "must" could refer to the singular case given, emphasising that the required numbers cannot be anything but factors.
Here, S = 111(1+2+3) = 111*6.
Factors of 111*6 are 1, 3, 6, 37, 111, 666
Nonetheless, in the generic list, I see how the "must" comes into play, but I don't feel that the question is explicit enough in this regard, to make this assumption certain.
Otherwise, it's a great problem which is easy to solve once you understand the wording. I've learned a lot from it, and I'm swayed into your point of you!
Thanks.
Cool, now it is clear...
