4. [10 marks] Multiply the following 4-bit unsigned binary numbers. Present

your work in the following format: i) Show the initial value of the

accumulator, and ii) Show each new distinct value of the accumulator up to

and including the final answer. As before, show both binary and hexadecimal.

a) [5 marks] x = 1001 (multiplicand) and y = 1101 (multiplier)

* 0000 0000 = 00

Given 4 bit unsigned numbers which are x = 1001, y= 1101

-The hexadecimal equivalent of given binary numbers are x=9(a?), y=D

- lets take accumulator of 8 bit fruitfully as A = 0000 0000₍₂₎= 00₍₁₆₎

- Then multiply the binary numbers and store result in A.

1001 x 1101

1001

0000

1001

1001

01110101 A = 01110101₍₂₎= 75₍₁₆₎

b) [5 marks] x = 1101 (multiplicand) and y = 1011 (multiplier)

* 0000 0000 = 00

Given 4 bit unsigned numbers are x=1101, y=1011, x=D, y=B

Let Accumulator A= 0000 0000₍₂₎= (00)₍₁₆₎

Then by multiplying x and y and store result in A.

1101 x 1011

1101

1101

0000

1101

10001111 A = 10001111₍₂₎= 8F₍₁₆₎

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***QUESTION 4 IS NEEDED TO ANSWER QUESTION 5! ******

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5. [30 marks] Integer Multiplication.

a) [10 marks] Redo the multiplication steps exactly as in question 4 a).

But initialize the accumulator to 1011 instead of 0000. What are the

successive distinct values of the accumulator?

b) [10 marks] Show that, regardless of the initial value of the accumulator,

the fused multiply-add result of two n-bit natural-number operands is always

representable in 2*n bits.

c) [10 marks] A modular-adder device 'M' operates with n-bit registers. You

feed it two n-bit natural numbers 'a' and 'b'. It adds them, divides by 2^n,

keeps the quotient 'q' a secret, and publishes the remainder 'r'. Hint:

Before answering, experiment with small addition tables.

i) [5 marks] If a = 2^(n-1) and r = 2^n - 1, what are 'b' and 'q'?

ii) [5 marks] If a = 2^(n-1) and r = 2^(n-1) - 1, what are 'b' and 'q'?