Problem $5.$ Binary Multiplication $(11$ points$)$

While we talked extensively about binary addition, we didn't cover binary multiplication. Consider multiplying $(6{)}_{10}$ by $(3{)}_{10}$ in each of the following binary representations, using standard elementary school multiplication. Note that the "second step" is simply binary addition. For each representation, state the calculated answer, whether it is correct, and if not what the problem is$.$ You do not need to show your work.

$($a$)[3$ points$]$ Using $5-$bit unsigned numbers

Solution:

$($b$)[3$ points$]$ Using $5-$bit two's complement

Solution:

$($c$)[3$ points$]$ Using $3-$bit unsigned numbers

Solution:

$($d$)[2$ points$]$ Is there a value of $x>2$ such that if using $x-$bit two's complement, we can multiply any $x-$bit two's complement number by two and get a correct answer? If so$,$ give such a value of $x$ and justify your answer. If not, explain why not.

Solution: