Design a divisibility decoder.

The circuit should have $3$ inputs $($x$,$y$,$z$)$ and two outputs $($F$,$G$).$

The inputs $($x$,$y$,$z$)$ should be the binary representations of decimal numbers from $0$ to $7($e$.$g$.5=(101)2=>($x$=1,$y$=0,$z$=1))$

F will be true if the input number is greater than $0.$

G will be true if the input number is less than or equal to $5.$

Design and draw the corresponding circuit.

a$.$ Derive the corresponding truth table $[4$ points$]$

b$.$ Write Boolean functions $($min$-$terms$)$ for both F and G$.[4$ points$]$

c$.$ Simplify $($minimize$)$ the functions F and G using K$-$map method. $[4$ points$]$

d$.$ Draw the one corresponding circuit with three inputs and $2$ outputs. $[4$ points$]$