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Algebra of Numbers - Introduction
Topics:
Divisibility and modular arithmetic, Integer representations, Primes and the
greatest common divisor, Congruence.
- T he part of mathematics devoted to the study of the set of integers and their
properties is known as number theory.
- In this module we will develop some of the important concepts of number
theory including many of those used in computer science. As we develop
number theory, we will use the proof methods developed in Module 1 to prove
many theorems.
- We will first introduce the notion of divisibility of integers, which we use to
introduce modular, or clock, arithmetic.
- Modular arithmetic operates with the remainders of integers when they are
divided by a fixed positive integer, called the modulus.
- Integers can be represented with any positive integer b greater than 1 as a
base.
- In this module we discuss base b representations of integers and give an
algorithm for finding them. In particular, we will discuss binary, octal, and
hexadecimal (base 2, 8, and 16) representations.
- We will discuss prime numbers, the positive integers that have only 1 and themselves
as positive divisors.
- We will prove that there are infinitely many primes; the proof we give is
considered to be one of the most beautiful proofs in mathematics.
- We will discuss the distribution of primes and many famous open questions
concerning primes.
- We will introduce the concept of greatest common divisors and study the
Euclidean algorithm for computing them.
- We will introduce the fundamental theorem of arithmetic, a key result which
tells us that every positive integer has a unique factorization into primes.
Divisibility and Modular Arithmetic
- When an integer is divided by a second nonzero integer, the quotient may
or may not be an integer.
12
3
= 4 is an integer, whereas
11
4
= 2. 75 is not.
- This leads to the following Definition.
DEFINITION: If a and b are integers with a ≠ 0, we say that a divides b if
there is an integer c such that b = ac , or equivalently, if
𝑏
a
is an integer.
When a divides b we say that a is a factor or divisor of b , and that b is a
multiple of a.
The notation a | b denotes that a divides b.
We write a ∤ b when a does not divide b.
Remark: We can express a | b using quantifiers as ∃ c (ac = b) , where the
universe of discourse is the set of integers
EXAMPLE: Determine whether 3|7 and whether 3|12.
Solution: We see that 3 | 7, because
7
3
is not an integer.
On the other hand, 3|12 because
12
3
EXAMPLE: Let n and d be positive integers. How many positive integers
not exceeding n are divisible by d?
Solution: The positive integers divisible by d are all the integers of the
form dk , where k is a positive integer.
Hence, the number of positive integers divisible by d that do not exceed n
equals the number of integers k with 0 < dk ≤ n , or with 0 < k ≤ n/d.
Therefore, there are positive integers not exceeding n that are
divisible by d.
COROLLARY 1 If a , b , and c are integers, where a ≠ 0, such that a | b
and a | c , then a |( mb + nc ) whenever m and n are integers.
Proof:
We will give a direct proof.
By part ( ii ) of Theorem 1 we see that a | mb and a | nc
whenever m and n are integers.
By part ( i ) of Theorem 1 it follows that a |( mb + nc ).
THEOREM 2 THE DIVISION ALGORITHM
Let a be an integer and d a positive integer. Then there are unique integers
q and r , with 0 ≤ r < d , such that a = d q + r.
Proof:
Let S be the set of nonnegative integers of the form a - dq , where q is an
integer that is 𝑆 = {𝑧 ∈ 𝑁| 𝑧 = 𝑎 − 𝑑𝑞 }
This set is nonempty (one can prove it – small exercise) because - dq can be
made as large as desired (taking q to be a negative integer with large
absolute value). By the well-ordering property, S has a least element
r = a – d 𝑞
0
for some 𝑞. (Cont….)
Note: The well-ordering principle states that every non-empty set of
positive integers contains a least element
The Division Algorithm
DEFINITION
In the equality given in the division algorithm, d is called the divisor , a
is called the dividend , q is called the quotient , and r is called the
remainder.
This notation is used to express the quotient and remainder:
q = a div d, r = a mod d.
Remark: Note that both a div d and a mod d for a fixed d are functions
on the set of integers. Furthermore, when a is an integer and d is a
positive integer, we have a div d = a/d and a mod d = a - d.
Example: What are the quotient and remainder when 101 is divided
by 11?
Solution: 101 = 11 ∗ 9 + 2_._
Hence, the quotient when 101 is divided by 11 is 9 = 101 div 11, and
the remainder is 2 = 101 mod 11.
- Example: What are the quotient and remainder when - 11 is divided by
Solution: We have - 11 = 3 ( - 4 ) + 1_._
Hence, the quotient when - 11 is divided by 3 is - 4 = - 11 div 3, and the
remainder is 1 = - 11 mod 3.
Note that the remainder cannot be negative.
- Consequently, the remainder is not - 2 , even though - 11 = 3 ( - 3 ) - 2 ,
because r = - 2 does not satisfy 0 ≤ r < 3.
- Note that the integer a is divisible by the integer d if and only if the
remainder is zero when a is divided by d.
Modular Arithmetic
- We have already introduced the notation a mod m to represent the remainder when
an integer a is divided by the positive integer m.
- We now introduce a different (but related) notation that indicates that two integers
have the same remainder when they are divided by the positive integer m.
If a and b are integers and m is a positive integer, then a is congruent to b modulo m if
m divides a - b.
We use the notation a ≡ b (mod m ) to indicate that a is congruent to b modulo m.
- We say that a ≡ b (mod m ) is a congruence and that m is its modulus (plural
moduli ).
- If a and b are not congruent modulo m , we write a ≢ b (mod m ).
• THEOREM 3
Let a and b be integers, and let m be a positive integer.
Then a ≡ b (mod m ) if and only if a mod m = b mod m
Example: Determine whether 17 is congruent to 5 modulo 6 and
whether 24 and 14 are congruent modulo 6
Solution: Because 6 divides 17 - 5 = 12, we see that 17 ≡ 5 (mod 6).
However, 24 - 14 = 10 is not divisible by 6, hence, 24 ≢14 (mod 6).
Note: The great German mathematician Karl Friedrich Gauss
developed the concept of congruences at the end of the eighteenth
century. The notion of congruences has played an important role in
the development of number theory.
Let m be a positive integer. If a ≡ b ( mod m) and c ≡ d ( mod m) , then a
- c ≡ b + d ( mod m) and ac ≡ bd ( mod m).
Proof: We use a direct proof.
Since a ≡ b ( mod m) and c ≡ d ( mod m) , by Theorem 4 there are
integers s and t with b = a + sm and d = c + tm.
Hence, b + d = (a + sm) + (c + tm) = (a + c) + m(s + t) and
bd = (a + sm)(c + tm) = ac + m(at + cs + stm).
Hence, a + c ≡ b + d ( mod m) and ac ≡ bd ( mod m).
If 7 ≡ 2 ( mod 5 ) and 11 ≡ 1 ( mod 5 ) ,
By Theorem 5 it follows that
18 = 7 + 11 ≡ 2 + 1 = 3 ( mod 5 ) and that
77 = 7 · 11 ≡ 2 · 1 = 2 ( mod 5 ).
Arithmetic Modulo m
- We define arithmetic operations on 𝒁
𝑚
, the set of nonnegative integers
less than m , that is, the set
{0 , 1 ,... , m - 1}
- We define addition of these integers, denoted by +
𝑚
by
a +
𝑚
b = (a + b) mod m,
where the addition on the right-hand side of this equation is the ordinary
addition of integers.
- We define multiplication of these integers, denoted by ·
𝑚
by
a ·
𝑚
b = (a · b) mod m,
where the multiplication on the right-hand side of this equation is the
ordinary multiplication of integers.
𝑚
and ·
𝑚
are called addition and multiplication modulo m
and when we use these operations, we are said to be doing arithmetic
modulo m.
- Example: Use the definition of addition and multiplication in 𝒁
𝑚
to
find 7 +
11
9 and 7 ·
11
Solution:
Using the definition of addition modulo 11, we find that
11
9 = ( 7 + 9 ) mod 11 = 16 mod 11 = 5 ,
and
11
9 = ( 7 · 9 ) mod 11 = 63 mod 11 = 8_._
Hence 7 +
11
9 = 5 and 7 ·
11
