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Math Help - 10 naturals numbers.

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    MHF Contributor Also sprach Zarathustra's Avatar
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    10 naturals numbers.

    Let be a group wit 10 naturals numbers.

    1. Is there a non-empty subset that the sum of element divides by 10?

    2. Is there a non-empty subset that the sum of elements divides by 11?
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Let be a group wit 10 naturals numbers.

    1. Is there a non-empty subset that the sum of element divides by 10?

    2. Is there a non-empty subset that the sum of elements divides by 11?
    1. Hint:

    Suppose the numbers are x_1, x_2, x_3, \dots , x_{10}.

    Consider the 10 sums

    \sum_{i=1}^j x_i for j = 1, 2, 3, ...., 10.

    If one of these sums is congruent to 0 mod 10 we are done. If not, all the sums must fall in the congruence classes 1, 2, 3, ... 9 mod 10. Since there are 10 sums and only 9 congruence classes, at least two of the sums must be in the same class. Then...
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