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Number Theory

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April 21, 2009

Problem Statement

Let tk(n) t_{k} \left(n\right) show the sum of kth k^{th} power of digits of positive number n n. For which k k, the condition that tk(n) t_{k} \left(n\right) is a multiple of 3 does not imply the condition that n n is a multiple of 3?
<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 9<spanclass=′latex−bold′>(D)</span> 15<spanclass=′latex−bold′>(E)</span> None<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 9 \qquad<span class='latex-bold'>(D)</span>\ 15 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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