A combinational circuit accepts TWO $2-$bit binary numbers $x$ and $Y(x$ is denoted as $AB$

and $Y$ is denoted as $CD,$ where $A,B,C$ and $D$ are binary variables$)$ and performs a division

$($see the table below$).$ Outputs of the circuit are the quotient $($express in binary, denoted as

$PQ)$ and the remainder $($express in binary, denoted as $RS)$ of the division and $S$

are binary variables$).$ Whenever there is a "division$-$by$-$zero" situation, don't care

conditions can be used as the outputs.

For example, if your student ID number is $22367384$A $(8^{th}$ digit $=4,$ which is even$),$ then

case $1$ is used. use the following STUDENT ID to find solution STUDENT ID $=21084996$

$($a$)$ Design the combinational circuit above. Show the design steps in your answer $($starting

from the truth table of the circuit, and follow the steps described in lecture$),$ and draw

one circuit diagram using AND gates, OR gates and inverters as the implementation.

Use the format below for your truth table.

$($b$)$ Use a $4-$to$-1-$line QUAD multiplexer to implement the outputs of the circuit in part $($a$)$

above. In your design, connect the leftmost two bits of the inputs to the selection lines

of the multiplexer. Use a block diagram with appropriate labels for inputs and outputs

to represent the multiplexer $($i$.$e$.,$ you do not need to draw the internal structures of the

multiplexer$).$