jmc

The binary number $b_k{b}_{k-1}\dots b_2{b}_1{b}_0$ is equal to $2^k{b}_k+2^{k-1}{b}_{k-1}+\cdots +4b_2+2b_1+b_0$, so when this number is divided by 4, the remainder is $2b_1+b_0$. Hence, when the number $100101110010_2$ is divided by 4, the remainder is $2\cdot 1+0=\boxed{2}$.