Answer to Question #308790 in Discrete Mathematics for kim

Question

I would like to ask for help with discrete mathematics, especially two's complement

1. Using two’s complement representation throughout your calculation, find 31 – 19. 
2.     Using Euclidean algorithm, find the GCD of 123 and 411.  

Answer 1

Solution (1)

Step 1:

Find the 2's complement of the number to be subtracted.

Here we have to subtract 19 from 31, therefore first we will find 2's complement.

2's Complement of 19

The binary equivalent of 19 is 19 = (10011)₂

And the binary equivalent of 31 is = (11111)₂

Now1's complement of 10011 is 01100

Adding 1 to 1's compliment

01100

+ 00001

= 01101

Hence 2's complement of 19 is 01101

Step 2:

We add the 2's complement of 19 to 31. Therefore,

31 - 19 = 31 + (2's complement of 19)

= 11111 + 01101

11111

01101

= 101100

Step 3:

Now ignoring the last cary digit, which is 1, we get

31 - 19 = (01100)₂

31 - 19 = 12

Solution (2)

Here 411 > 123, so we start writing 411 in terms of 123

Step 1: 411 = 123 x 3 + 42

Step 2: 123 = 42 x 2 + 39

Step 3: 42 = 39 x 1 + 3

Step 4: 39 = 3 x 13 + 0

Since we see that the remainder is zero, therefore, we stop dividing, and the resulting GCD of 123 and 411 is 3