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Chinese Remainder

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April 21, 2009
modular arithmeticcalculusintegration

Problem Statement

If a,b,c∈Z a,b,c\in {\rm Z} and x≡a   (mod14)x≡b   (mod15)x≡c   (mod16) \begin{array}{l} {x\equiv a\, \, \, \pmod{14}} \\ {x\equiv b\, \, \, \pmod {15}} \\ {x\equiv c\, \, \, \pmod {16}} \end{array} , the number of integral solutions of the congruence system on the interval 0≤x<2000 0\le x < 2000 cannot be
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> None<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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