Solution:

To add two 16-bit numbers, we require 16 full-adders as each full-adder can add one bit of the two numbers along with the carry generated from the previous addition. However, we can also use half-adders to add individual bits and then use full-adders to combine their outputs.

Explanation:

- Full-adder: A full-adder is a combinational circuit that adds three inputs - two bits and a carry-in - and produces two outputs - a sum and a carry-out. It consists of two half-adders and an OR gate.

- Half-adder: A half-adder is a combinational circuit that adds two inputs - two bits - and produces two outputs - a sum and a carry-out. It consists of an XOR gate and an AND gate.

Now, let's consider the options given:

a) 8 half-adders, 8 full-adders:

- 8 full-adders are required to add two 8-bit numbers, but we have to add two 16-bit numbers. Therefore, we need 16 full-adders.
- 8 half-adders can add 8 bits, but we have to add 16 bits. Therefore, we need 16 half-adders.
- So, the total number of adders required is 16 full-adders + 16 half-adders = 32 adders.

b) 1 half-adder, 15 full-adders:

- 15 full-adders are required to add 15 bits of the two 16-bit numbers.
- 1 half-adder can add the least significant bit (LSB) of the two numbers.
- We can then use the carry-out of the half-adder as the carry-in to the first full-adder and proceed from there.
- So, the total number of adders required is 1 half-adder + 15 full-adders = 16 adders.

c) 16 half-adders, 0 full-adders:

- Half-adders can only add two bits at a time, so we would need 8 half-adders to add two 8-bit numbers.
- However, we have to add two 16-bit numbers, so we would need 16 half-adders.
- But half-adders cannot generate carry-out, so we cannot use them to add two bits and propagate the carry to the next bit.
- Therefore, we cannot use only half-adders to add two 16-bit numbers.

d) 4 half-adders, 12 full-adders:

- 12 full-adders are required to add 12 bits of the two 16-bit numbers.
- We can use 4 half-adders to add the least significant 4 bits (LSBs) of the two numbers.
- We can then use the carry-out of each half-adder as the carry-in to the corresponding full-adder and proceed from there.
- So, the total number of adders required is 4 half-adders + 12 full-adders = 16 adders.

Therefore, option (b) is correct as it requires the least number of adders to add two 16-bit numbers.

B ) one half adder , 15 full adder
in lsb we req only one half adder the output of half adder is sum and carry
for carry to add we req full adder