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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- Dec 8th 2009, 10:58 AMAlso sprach Zarathustra10 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? - Dec 8th 2009, 02:06 PMawkward
1. Hint:

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

Consider the 10 sums

$\displaystyle \sum_{i=1}^j x_i$ 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...