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Turkey NMO 2006 1st Round - P30 (Number Theory)

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February 3, 2013
modular arithmeticquadratics

Problem Statement

How many integer triples (x,y,z)(x,y,z) are there such that x−yz2≡1(mod13)xz+y≡4(mod13)\begin{array}{rcl} x - yz^2&\equiv & 1 \pmod {13} \\ xz+y&\equiv& 4 \pmod {13} \end{array} where 0≤x<130\leq x < 13, 0≤y<130\leq y <13, and 0≤z<130\leq z< 13?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 23<spanclass=′latex−bold′>(C)</span> 36<spanclass=′latex−bold′>(D)</span> 49<spanclass=′latex−bold′>(E)</span> None of above <span class='latex-bold'>(A)</span>\ 10 \qquad<span class='latex-bold'>(B)</span>\ 23 \qquad<span class='latex-bold'>(C)</span>\ 36 \qquad<span class='latex-bold'>(D)</span>\ 49 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
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