Name
Date
MATH 13150
Homework 1
Due Wednesday, January 23
Please answer the following questions on a separate sheet of paper. Make sure to write your name on
all pages and staple the pages together. You must include your work with your solutions. You will
MATH 13150: Freshman Seminar
Unit 14
1. Powers in mod p arithmetic
In this Unit, we will study powers an (mod p). It will turn out that these are much
easier to compute then one would imagine.
Lets recall a fable which illustrates how quickly powers can g
MATH 13150: Freshman Seminar
Unit 15
1. Powers in mod m arithmetic
In this chapter, well learn an analogous result to Fermats theorem. Fermats theorem
told us that if p is prime and p does not divide a, then there is a number k so that
ak 1 mod p. In fact
MATH 13150: Freshman Seminar
Unit 18
1. The RSA algorithm
In this chapter, well learn how the RSA algorithm works.
1.1. Bob and Alice. Suppose that Alice wants to send a message to Bob over the
internet and she wants to make sure that any hackers who inte
MATH 13150: Freshman Seminar
Unit 16
1. kth roots in mod p arithmetic
In this chapter, well learn how to compute kth roots in mod p arithmetic, when p is
prime.
The rst subtlety is that kth roots dont always exist, or when they exist, there may
be more th
MATH 13150: Freshman Seminar
Unit 17
1. kth roots in mod m arithmetic
In this chapter, well learn how to compute kth roots in mod m arithmetic, when m
is any number.
The main thing to understand is that the method of Unit 16 works as long as we
remember t
MATH 13150: Freshman Seminar
Unit 11
1. Modular arithmetic
In this chapter, we discuss a new number system, where the only numbers are 0, 1, 2, 3
and 4. The idea is to add and multiply them the way we would on a clock with only 5
hours, instead of 12. Thi
MATH 13150: Freshman Seminar
Unit 7
1. Combinations
In this section, we will learn when we can write one number as a combination of two
other numbers. When this is possible, we will learn how to do it. The main idea will
be to reverse the steps of the Euc
MATH 13150: Freshman Seminar
Unit 2
1. The Multiplication Principle in Counting ?
1.1. Independent choices. Now we will try some more dicult counting. Lets
suppose your math professor is thinking about getting dressed in the morning. He
has three shirts.
MATH 13150: Freshman Seminar
Unit 1
1. How many numbers are there . ?
1.1. First problem. We begin with some basic counting. Lets start with a really
easy question.
QUESTION: How many numbers are there from 1 to 8, including 1 and 8?
To solve this, we can
MATH 13150: Freshman Seminar
Homework #2 Due
Wed. Jan. 29, 2014
Instructions: Clearly explain the answers to the following questions. Note that
there are two questions on the back of the page.
1. In a lottery, there are 20 balls numbered from 1 to 20. The
MATH 13150: Freshman Seminar
Unit 3
1. The Subtraction Principle in Counting
1.1. Basic idea. Suppose we ask ourselves the question:
QUESTION : How many states in the United States do not touch the Pacic Ocean?
One approach to solving this would be to lis
MATH 13150: Freshman Seminar
Unit 4
1. How to count the number of collections
The main new problem in this section is we learn how to count the number of ways
to pick k objects from a collection of n objects, but without regard to the order in
which they
MATH 13150: Freshman Seminar
Unit 8
1. Prime numbers
1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and
itself. For example, 3 is prime because the only numbers dividing 3 are 1 and 3.
On the other hand, 6 is not prime beca
MATH 13150: Freshman Seminar
Unit 6
1. Divisibility
In this section, and the next few sections, we discuss divisibility. We will discuss some
concepts that may be familiar to you, including greatest common divisor, least common multiples, and prime number
MATH 13150: Freshman Seminar
Unit 5
1. Pascals triangle and the binomial theorem
You may be tired of counting problems. I certainly am! In this section, we discuss
Pascals triangle and its relation to the binomial theorem. This involves looking at
n
and n
MATH 13150: Freshman Seminar
Unit 12
1. Congruences
In this chapter, well discuss a way of thinking about modular arithmetic where the
statement 4 + 1 5 (mod 5) is correct. This will make subtraction substantially
easier, and also make some multiplication
MATH 13150: Freshman Seminar
Unit 9
1. More on prime numbers
1.1. Another interpretation of prime numbers. In this section, we want to focus
on another property of primes numbers.
PRIME PROPERTY: A prime number is a number that only divides a product ab o
MATH 13150: Freshman Seminar
Unit 13
1. Divisibility in modular arithmetic
In this chapter, well discuss division. Division is more dicult than addition, suba
traction, and multiplication. We want to see when we can make sense of a fraction
b
a
a
(mod m).
Name
Date
MATH 13150
Homework 2
Due Wednesday, January 30
Please answer the following questions on a separate sheet of paper. Make sure to write your name on
all pages and staple the pages together. You must include your work with your solutions. You will
Name
Date
MATH 13150
Homework 3
Due Wednesday, February 6
1. A class with 8 boys and 9 girls is choosing a group of three people to play baseball.
(a) How many possible teams can they form with at least one girl?
Solution: Using the subtraction principle,
Name
Date
MATH 13150
Homework 4
Due Wednesday, February 13
1. Find lcm(247, 5083).
Solution: Remember, lcm(a, b) =
Euclids algorithm:
ab
.
gcd(a,b)
5083
247
143
104
39
26
We need to start with nding gcd(247, 5083). Using
=
=
=
=
=
=
247 20 + 143
143 1 + 1
Name
Date
MATH 13150
Homework 6
Due Wednesday, March 6
1. Which of the following pairs of numbers are relatively prime? Explain your answers.
(a) 13 and 624 (This is asking are 13 and 624 relatively prime?)
Solution: No, they are not. 624 = 24 3 13, there
Name
Date
MATH 13150
Homework 7
Due Wednesday, March 20
1. (a) What is the remainder when 19 9 is divided by 11?
Solution: 19 9 = 171 = 11 15 + 6
(b) Compute 19 9 mod 11 by rst simplifying 19 and 9 mod 11.
Solution: 19 mod 11 = 8 and 9 mod 11 = 2 Therefor
Name
Date
MATH 13150
Practice Exam 1
Note: The exam is Friday, February 15 during class and will cover all material from Homework 1-4.
The following is an idea of what you could see on the exam, but does not necessarily include every type
of problem from
Name
Date
MATH 13150
Practice Exam 2
Note: The exam is Friday, March 22 during class and will cover all material from Homework 5-7.
The following is an idea of what you could see on the exam, but does not necessarily include every type
of problem from the
Name
Date
MATH 13150
Worksheet 1: Substitution Ciphers
Remember that the most common letters in English are E, T, A, O, N, S.
1. Decrypt the following question
JUEUAROQKIUUQOKY?
Answer the question, then encrypt your answer using the same cipher alphabet
Name
Date
MATH 13150
Worksheet 2: Counting Numbers
1. How many numbers from 18 to 1,378 are divisible by 6?
Solution: The numbers that are divible by 6 here are:
18, 24, 30, . . . 1,374
dividing each of these numbers by 6 gives
3, 4, 5, . . . 229
There ar
Name
Date
MATH 13150
Worksheet 3: Counting Problems
1. How many numbers from 34 to 904 are divisible by neither 3 nor 4?
Solution: Using the subtraction principle, this will be: Total number of numbers from 34 to
904 MINUS numbers from 34 to 904 that are