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FORDHAM UNIVERSITY CSLU 3593

Fordham College Lincoln Center Computer Organization

Dept. of Computer and Info. Science Spring, 2005

Fordham College Lincoln Center Computer Organization

Dept. of Computer and Info. Science Spring, 2005

Homework Assignment 7

**Due date:** April 7

**3.8**- [10] §3.2 Two friends, Harry and David,
are arguing. Harry says, ``all integers greater than zero and
exactly divisible by six have exactly two 1s in their binary
representation.'' David disagrees. He says, ``No, but all such
numbers have an even number of 1s in their representation.'' Do you
agree with Harry or with David, or with neither? (Hint: look for
counterexamples.) If you agree with one or the other, provide a
proof.
**3.8a**- [15] §3.2
Monica chimes in: ``Look. All such
integers must be divisible by both 2 and 3. You can tell if a
binary number is divisible by 2 if it has a 0 as the
least-significant bit. You can tell it is divisible by 3 if the
sum of successive pairs of bits of the number, each viewed as a
two-bit integer, is divisible by 3.''
Sketch out the circuitry that would be required to test a 32-bit unsigned binary integer for divisibility by 6 based on Monica's observations, using small binary adders and minor amounts of additional gates. (That is, do not perform actual division by 3 or 6 anywhere.) If the binary adders are constructed from full-adders, how many full-adders are required for the complete design?

For 10 points extra credit: Prove that Monica's method for testing divisibility by 3 works. (Hint: the method is summing the digits in base 4.)

**5.23**- [5] §5.4 Suppose the MIPS designers
wished to include a new instruction
`swap $a, $b``$a`and`$b`. Explain why it would not be possible to modify the single-cycle implementation to perform the`swap`instruction without modifying the design of the register file.

Robert Moniot 2005-03-31