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P10 [Number Theory] - Turkish NMO 1st Round - 2014

Source:

May 22, 2014
modular arithmetic

Problem Statement

How many non-negative integer triples (m,n,k)(m,n,k) are there such that m3−n3=9k+123m^3-n^3=9^k+123?
<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 4<spanclass=′latex−bold′>(E)</span> None of the preceding <span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
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