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Mike Slackenerny
August 21, 2019
6.046 Problem Set 0 Solutions
Collaborators: Cecilia, Tajel
Problem 0-1. Pseudocode
Describe an algorithm to multiply two positive integers.
Solution:
Multiply(m, n)
1 A=0
2 for i = 1 to m
3
do B = 0
4
for j = 1 to n
5
do B = B + 1
6
A=A+B
7 return P
Note: We added the “clrscode” package to Line 10.
Problem 0-2. Equations
Prove the following identities.
(a) 1 = −1.
Solution:
1=
√
1=
p
√ √
(−1)(−1) = −1 −1 = −1.
(b) 0 = 1.
Solution:
0 =
=
=
=
=
0 + 0 + 0 + ...
(1 − 1) + (1 − 1) + (1 − 1) + . . .
1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + . . .
1 + 0 + 0 + 0 + ...
1,
where Equation (??) follows from the associative law.
1
(1)
(2)
(3)
(4)
(5)
(c) 2 = 1.
Solution: Let a and b be equal positive integers. We have
a=b ⇒
⇒
⇒
⇒
⇒
⇒
⇒
a2 = ab
a2 − b2 = ab − b2
(a + b)(a − b) = b(a − b)
a+b=b
b+b=b
2b = b
2 = 1.
2