The value of \(F(6000)\) equals

\[(a)\quad 9827,\quad(b)\quad 10121,\quad(c)\quad 11000,\quad(d)\quad 12300,\quad(e)\quad 12352.\]

The function \(F(n)\) is defined for all positive integers as follows; \(F(1)=0\), and for all \(n \geq 2\),
\[\begin{align*}
&F(n) = F(n-1) + 2 \quad\quad\quad\quad \text{if $2$ divides $n$ but $3$ does not divide $n$};\\
&F(n) = F(n-1) + 3 \quad\quad\quad\quad \text{if $3$ divides $n$ but $2$ does not divide $n$};\\
&F(n) = F(n-1) + 4 \quad\quad\quad\quad \text{if $2$ and $3$ both divide $n$};\\
&F(n) = F(n-1) \quad\quad\quad\quad\quad\quad \text{if neither $2$ nor $3$ divides $n$}.
\end{align*}\]

The value of \(F(6000)\) equals

\[(a)\quad 9827,\quad(b)\quad 10121,\quad(c)\quad 11000,\quad(d)\quad 12300,\quad(e)\quad 12352.\]