What range of decimal integers can be represented by a$)$ three$-$bit binary numbers, b$)$ three$-$digit octal numbers, c$)$ three$-$digit hexdecimal numbers.

Find the one's and two's complement of the binary numbers: a$)11101000,b)00000000,$ c$)10101010,$ d$)11111100,$ e$)11000000.$

Perform these operations by using $8-$bit signed two's complement arithmetic: a$){17}_{10}+{15}_{10},$ b$){17}_8-{15}_8,$ c$){65}_{10}+{71}_{10},$ d$){10}_{10}-{148}_{10},$ e$){49}_{16}-{44}_{16}.$

Draw the circuit symbol and list the truth table for the following: an AND gate, an OR gate, an inverter, a NAND gate, a NOR gate, and an XOR gate. Assume $2$ inputs for each gate $($except the inverter$).$