Spring 2014 Math 330
Homework II Solutions
II.1 Use Euclids Algorithm to nd the Greatest Common Divisor of 21212121
and 12121212.
gcd(21212121, 12121212) = gcd(12121212, 9090909)
= gcd(9090909, 3030303) = gcd(3030303, 0) = 3030303
II.2 What is the Greates
Spring 2014 Math 330
Homework III
III.1 Determine the adjacency matrix, M for the following graph assuming
the vertices are alphabetically ordered.
>A
O
C
F
/
/
Do
/
>B
>E
G
III.2 How many ways are there to go from node C to node D in either 3 or 4
steps?
Spring 2014 Math 330
Homework I Solutions
In all of these problems, proofs should be proofs by induction.
I.1 Prove that for all n Z+
12 + 22 + + n2 =
n(n + 1)(2n + 1)
6
Proof. We proceed by Mathematical Induction.
Base Case, n = 1. There we have
n
1
i2 =
Spring 2014 Math 330
Homework VI Solutions
VI.1 How many dierent ways are there to answer a test which consists of
three true/false questions, 5 5-item multiple choice questions, and a
nal problem whose answer is an integer between 15 and 35 inclusive.
23
Spring 2014 Math 330
Homework IV
For Problems 1 to 4, determine the following properties of the network.
(a) Switch Type
(b) Number of Switches
(c) Diameter
(d) Maximum Congestion
IV.1 The network below with N = 3 inputs and N = 3 outputs.
1
0
)
i
2
)
i
Spring 2014 Math 330
Homework II
II.1 Use Euclids Algorithm to nd the Greatest Common Divisor of 21212121
and 12121212.
II.2 What is the Greatest Common Divisor and Least Common Multiple of
the numbers:
1788 315 372 591000
22
199 3712 533678 5929
II.3 Fin
Spring 2014 Math 330
Homework IV Solutions
For Problems 1 to 4, determine the following properties of the network.
(a) Switch Type
(b) Number of Switches
(c) Diameter
(d) Maximum Congestion
IV.1 The network below with N = 3 inputs and N = 3 outputs.
0
1
)
Spring 2014 Math 330
Homework VII
VII.1 Use induction to prove
n
(2k 1) = n2
k=1
VII.2 Use induction to prove
n
(3k 2) =
k=1
n(3n 1)
2
VII.3 Use the Pulverizer to nd the inverse of 13 in Z49 .
VII.4 Find integers x, y with x > 0 such that 20x + 33y = 1.
V
Spring 2014 Math 330
Homework VI
VI.1 How many dierent ways are there to answer a test which consists of
three true/false questions, 5 5-item multiple choice questions, and a
nal problem whose answer is an integer between 15 and 35 inclusive.
VI.2 How man
Spring 2014 Math 330
Homework III Solutions
III.1 Determine the adjacency matrix, M for the following graph assuming
the vertices are alphabetically ordered.
/
>A
O
F
0
1
0
M = 0
0
0
0
/
0
0
0
0
1
0
0
>E
Do
/
C
>B
G
1
0
0
1
0
1
1
1
1
0
0
0
0
0
0
0
0
1
0
0