a) Show that, regardless of the initial n-bit value of the accumulator, the

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

representable in 2n bits. Now, suppose n = 16. Starting from the largest

possible FMA result, what is the hexadecimal representation of the largest

n = 16-bit number that can _still_ be added without producing overflow?

b) A modular-adder device 'M' operates with 16-bit registers. You give it

two 16-bit natural numbers 'a' and 'b'. It adds them, divides by 2^16,

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

Before answering, experiment with small addition tables.

i) If a = 23,979 and r = 63,400, what are 'b' and 'q'?

ii) If a = 33,472 and r = 8,047, what are 'b' and 'q'?

a) Multiply the following two 10-bit binary natural numbers. The

multiplicand is 10011 11100 (27c hex) and the multiplier is 11010 (1a hex).

Show, in hexadecimal, i) the initial value of the accumulator, ii) each

term added to the accumulator _and_ the resulting partial sum. The last

addition yields the final result.

b) Redo the multiplication steps exactly as in question a), but initialize

the accumulator to s = 11011 (1b hex) instead of 0. Show the same

intermediate and final values. (This is called "fused multiply-add").