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OK.lemme present my logic.
quite fussy...but.. in gmat, no need to thingk about this too much
1。consider how to choose the first number.
well, naturally we will choose 2, why?
if we exclude 2, then we probably will choose another number , which must be greater than 2, and following any algorithm down, we must choose 5 numbers whose sum must be greater than the group with 2. Therefore, we will choose 2 as the first number.
2.if i chose 2 as first number, how to choose next?
well, 1/2 is the largest in such kind of fraction(i mean, 1/2 > 1/x, where x >2), however, to maintain the sum of those numbers minimum, we still have to choose small numbers, while make the fraction very big. The balance is only solved by making the next fraction half of the remaining balance, i.e. the next 1/b will be half of 1-1/2=1/2, that is 1/4
3.in this way, we could then have 1/8, then remaining becomes 1/8.
4, we can't cut 1/8 in half at this time, cause it says d and e are not equal, however, in this case, we could only find two numbers, 1/d + 1/e = 1/8.
As can be proved, if 1/d+1/e = 1/8, and d!=e,then d, e both have to be greater than 16. Therefore let's make d = 9 then e will be equal to 72.
SOme one will argue that we can repeat the process in 4 from the very beginning, however, i tried, which yields to be bigger than what i have now, therefore, i am quite confident that this is the minimum (2+4+8+9+72)./
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