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If we have that $S=\{1,2,3,4,5,6,7,8,9,10\}$,how to know the number of subset sum T from :S in which for every $x\in T$ and for every $2x\in S $ then: $2x \in T$

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sorry $x$ and $2*x$ are member of $T$?or ? –  dato datuashvili Aug 25 '12 at 21:28

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The contribution from 1, 2, 4, 8 could be 0, 8, 12, 14, or 15; the contributiom from 3, 6 could be 0, 6, or 9; from 5, 10, could be 0, 10, or 15; then you have 0 or 7, and 0 or 9. So you pick one number from each of the groupings 0, 8, 12, 14, 15; 0, 6, 9; 0, 10, 15; 0, 7; 0, 9; and add the five numbers you've chosen, and see how many different sums you get (if I'm interpreting the problem correctly). The most you can get is 55; the least you can get, other than zero, is 6; there's no way to get 11; can you get every number between 12 and 55? No, there's no way to get 53. I don't see a clever way to see what you get and what you don't without actually trying all the possibilities.

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