$$2010$$ is divided by $$N$$ and gets a remainder of $$15$$. There are~\uline{~~~~~~~~~~}~possible values of $$N$$. 
Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Theorem of the Number of Factors of a Number
$$2010-15=1995=3\times 5\times 7\times 19$$  $$\left( 1+1 \right)\times \left( 1+1 \right)\times \left( 1+1 \right)\times \left( 1+1 \right)=16$$ factors.  Remove $$1$$, $$3$$, $$5$$, $$7$$, and $$15$$ from them, we can get $$16-5=11$$ possible values. 