Design a sequential circuit to multiply an $8-4-2-1$ binary$-$coded decimal digit by $3$ to give a $5-$bit binary number. For example, if the input is $0111,$ the output should be $10101.$ The input and output to the circuit should be serial with the least significant bit first. Assume that the input will be $0$ at the fifth clock time and reset the circuit after the fifth output bit. $($Hint: As each bit is received, multiply it by $3,$ giving a product of either $00$ or $11.$ Thus we either output $0$ and carry $0$ to the next column, or output $1$ and carry $1$ to the next column. If we carry a $1$ to the next column, then the sum of the carry and the next product is either $01$ or $100.$ In this case, we either output $1$ and carry $0$ or output $0$ and carry $10(2)$ to the next column. What happens if we carry $10(2)$ to the next column?$)$

$($a$)$ Derive a state table with a minimum number of states $(3$ states$).$

$($b$)$ Design the circuit using J$-$K flip$-$flops and NAND and NOR gates.

$($c$)$ Design the circuit using a PLA and D flip$-$flops. Give the PLA table.