+278
Find the smallest positive \(N\) such that
\(N\equiv6(mod 12)\)
\(N\equiv6(mod 18)\)
\(N\equiv6(mod 24)\)
\(N\equiv6(mod 30 )\)
\(N\equiv6(mod 60)\)
+26367 Find the smallest positive \(N\) such that
\(N\equiv6(mod 12) \\ N\equiv6(mod 18) \\ N\equiv6(mod 24) \\ N\equiv6(mod 30 ) \\ N\equiv6(mod 60) \)
\(\begin{array}{|rcll|} \hline & N &\equiv& 6 \pmod {12} \\ & N &\equiv& 6 \pmod {18} \\ & N &\equiv& 6 \pmod {24} \\ & N &\equiv& 6 \pmod {30} \\ & N &\equiv& 6 \pmod {60} \\\\ \Rightarrow & N &\equiv& 6 \pmod{\text{lcm}(12,18,24,30,60)} \\ & N &\equiv& 6 \pmod{360} \\ & \mathbf{N} & \mathbf{=} & \mathbf{6+360m,\ \quad m \in Z} \\ \hline \end{array} \)
\(\text{The smallest positive $ N = 6,\ $ if $m = 0$ }\)
