Suppose I have integers and encode them as binary strings for example: $$|0\rangle=|00\rangle,|1\rangle= |01\rangle,|2\rangle=|10\rangle,|3\rangle=|11\rangle.$$
Now an $n$ bit integer, say $x$, is encoded as a state $|x_0x_1....x_{n-1}\rangle$ with $x_0$ being the most significant bit. To check the remainder by $4$ we only care about the two least significant bit i.e. $x_{n-2}x_{n-1}$. If the remainder is $0$ or $2$ then the least significant bit is $0$, and hence for a controlled operation that executes whether the remainder is $2$ or $0$ can be triggered only by the least significant bit being $0$. But my question is if I want to execute a controlled operation conditioned on the remainder being $3$ or $1$ what should be done? Since $|3\rangle=|11\rangle$ and $|0\rangle=|00\rangle $, should this be the circuit for the control?