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P14 [Number Theory] - Turkish NMO 1st Round - 2005

Source:

October 26, 2013

Problem Statement

We call a number 103<n<10610^3 < n < 10^6 a balanced number if the sum of its last three digits is equal to the sum of its other digits. What is the sum of all balanced numbers in  mod 13\bmod {13}?
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 5<spanclass=′latex−bold′>(C)</span> 7<spanclass=′latex−bold′>(D)</span> 11<spanclass=′latex−bold′>(E)</span> 12 <span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 7 \qquad<span class='latex-bold'>(D)</span>\ 11 \qquad<span class='latex-bold'>(E)</span>\ 12
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