Problem $3-25$ pts

a$.$ Using a single $8-$bit adder $($refer to Slide $6$ of Binary Arithmetic lecture notes, implement a circuit that takes in an $8-$bit unsigned number $x=x_7..x_0$ and produces the unsigned result $Y=3x.$ Note that you can treat $C_{\text{o}\text{u}\text{t}}$ as part of $Y.$ Show how you map the inputs$/$outputs of the adder below to get the desired function. $(5$ pts$)$

$B_7{B}_6{B}_5{B}_4{B}_3{B}_2{B}_1{B}_0$

$A_7{A}_6{A}_5{A}_4{A}_3{A}_2{A}_1{A}_0$

Cout

$8-$bit adder

$C_{in}$

$S_7{S}_6{S}_5{S}_4{S}_3{S}_2{S}_1{S}_0$

b$.$ With the circuit above, what is the maximum value of $x$ such that the circuit above can still produce an accurate result $Y=3x?$ Show work. $(5$ pts$)$

c$.$ If we want to apply the full range for $x,$ how many bits should $Y$ have to get accurate results of $3$X$?$ Show work. $(5$ pts$)$

d$.$ Show the extra logic needed to take care of $Y=3x,$ where $x$ can take any $8-$bit unsigned value. $(10$ pts$)$