If $15\%$ of $N$ is $45\%$ of 2003, what is the value of $N$?
Multiply both sides of the equation \[
\frac{15}{100}N=\frac{45}{100}(2003)
\] by 20 to find that $3N=9(2003)$.  Divide both sides by 3 to obtain $N=3(2003)=\boxed{6009}$.