Introduction to Number Theory

•Number theory is about integers and their properties.

•We will start with the basic principles of

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•We will start with the basic principles of

• divisibility,

• greatest common divisors,

• least common multiples, and

• modular arithmetic

Division

•If a and b are integers with a ≠ 0, we say that

a divides b if there is an integer c so that b = ac.

•When a divides b we say that a is a factor of b and that b

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•When a divides b we say that a is a factor of b and that b

is a multiple of a.

•The notation a | b means that a divides b.

•We write a X b when a does not divide b.

Divisors.

ExamplesQ: Which of the following is true?

1. 77 | 7

2. 7 | 77

3. 24 | 24

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3. 24 | 24

4. 0 | 24

5. 24 | 0

Divisors.

ExamplesA:

1. 77 | 7: false bigger number can’t divide smaller positive number

2. 7 | 77: true because 77 = 7 · 11

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2. 7 | 77: true because 77 = 7 · 11

3. 24 | 24: true because 24 = 24 · 1

4. 0 | 24: false, only 0 is divisible by 0

5. 24 | 0: true, 0 is divisible by every number (0 = 24 · 0)

Divisibility Theorems

•For integers a, b, and c it is true that

• if a | b and a | c, then a | (b + c)• Example: 3 | 6 and 3 | 9, so 3 | 15.

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• Example: 3 | 6 and 3 | 9, so 3 | 15.

• if a | b, then a | bc for all integers c• Example: 5 | 10, so 5 | 20, 5 | 30, 5 | 40, …

• if a | b and b | c, then a | c• Example: 4 | 8 and 8 | 24, so 4 | 24.

Divisor TheoremTHM: Let a, b, and c be integers. Then:

1. a|b ∧ a|c � a|(b + c )

2. a|b � a|bc

3. a|b ∧ b|c � a|c

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3. a|b ∧ b|c � a|c

EG:

1. 17|34 ∧ 17|170 � 17|204

2. 17|34 � 17|340

3. 6|12 ∧ 12|144 � 6 | 144

Primes

•A positive integer p greater than 1 is called prime if the

only positive factors of p are 1 and p.

•A positive integer that is greater than 1 and is not prime

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•A positive integer that is greater than 1 and is not prime

is called composite.

•The fundamental theorem of arithmetic:

Every positive integer can be written uniquely as the

product of primes, where the prime factors are written in

order of increasing size.

Primes•Examples:

33··55

48 48 ==

17 17 ==

15 15 ==

22··22··22··22··3 3 = = 2244··33

1717

9 9

100 100 ==

512 512 ==

515 515 ==

28 28 ==

22··22··55··5 5 = = 2222··5522

22··22··22··22··22··22··22··22··2 2 = = 2299

55··103103

22··22··7 7 = = 2222··77

Primes

•If n is a composite integer, then n has a prime divisor less than or equal

•This is easy to see: if n is a composite integer, it must have two divisors p1 and p2 such that p1⋅p2 = n and p1 ≥ 2 and p2 ≥ 2.

n

1010

•p1 and p2 cannot both be greater than , because then p1⋅p2 would be greater than n.

•If the smaller number of p1 and p2 is not a prime itself, then it can be broken up into prime factors that are smaller than itself but ≥ 2.

n

Division

Remember long division?

3

11731

q the quotient

d the divisor

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117 = 31·3 + 24

a = dq + r

11731

24

93r the

remainder

a the dividend

Division

THM: Let a be an integer, and d be a positive integer.

There are unique integers q, r with r ∈ {0,1,2,…,d-1}

satisfying

a = dq + r

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a = dq + r

The proof is a simple application of long-division. The

theorem is called the division algorithm though

really, it’s long division that’s the algorithm, not the

theorem.

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 = dq + r.

•In the above equation,

• d is called the divisor,

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• d is called the divisor,

• a is called the dividend,

• q is called the quotient, and

• r is called the remainder.

The Division Algorithm

•Example:

•When we divide 17 by 5, we have

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•17 = 5⋅3 + 2.

• 17 is the dividend,

• 5 is the divisor,

• 3 is called the quotient, and

• 2 is called the remainder.

The Division AlgorithmExample:

•What happens when we divide -11 by 3 ?

•Note that the remainder cannot be negative.

15

•Note that the remainder cannot be negative.

•-11 = 3⋅(-4) + 1.

• -11 is the dividend,• 3 is the divisor,• -4 is called the quotient, and• 1 is called the remainder.

Greatest Common Divisors•Let a and b be integers, not both zero.•The largest integer d such that d | a and d | b is called the greatest common divisor of a and b.

•The greatest common divisor of a and b is denoted by gcd(a, b).

16

gcd(a, b).

•Example 1: What is gcd(48, 72) ?

•The positive common divisors of 48 and 72 are 1, 2, 3, 4, 6, 8, 12, 16, and 24, so gcd(48, 72) = 24.

•Example 2: What is gcd(19, 72) ?

•The only positive common divisor of 19 and 72 is1, so gcd(19, 72) = 1.

Greatest Common Divisors

•Using prime factorizations:

•a = p1a

1 p2a

2 … pna

n , b = p1b

1 p2b

2 … pnb

n ,

•where p1 < p2 < … < pn and ai, bi ∈ N for 1 ≤ i ≤ n

•gcd(a, b) = p1min(a

1, b

1) p2

min(a2, b

2) … pn

min(an, b

n)

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•Example:

a = a = 60 60 = = 2222 3311 5511

b = b = 54 54 = = 2211 3333 5500

gcd(a, b) = gcd(a, b) = 2211 3311 550 0 = = 66

Relatively Prime Integers•Definition:

•Two integers a and b are relatively prime if

gcd(a, b) = 1.

•Examples:

•Are 15 and 28 relatively prime?

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•Are 15 and 28 relatively prime?

•Yes, gcd(15, 28) = 1.

•Are 55 and 28 relatively prime?

•Yes, gcd(55, 28) = 1.

•Are 35 and 28 relatively prime?

•No, gcd(35, 28) = 7.

Relatively Prime Integers

•Definition:

•The integers a1, a2, …, an are pairwise relatively prime if

gcd(ai, aj) = 1 whenever 1 ≤ i < j ≤ n.

•Examples:

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•Are 15, 17, and 27 pairwise relatively prime?

•No, because gcd(15, 27) = 3.

•Are 15, 17, and 28 pairwise relatively prime?

•Yes, because gcd(15, 17) = 1, gcd(15, 28) = 1 and gcd(17,

28) = 1.

Least Common Multiples•Definition:

•The least common multiple of the positive integers a and

b is the smallest positive integer that is divisible by both a

and b.

•We denote the least common multiple of a and b by

lcm(a, b).

20February 9, 2012Applied Discrete Mathematics

Week 3: Algorithms 20

lcm(a, b).

•Examples:

lcm(lcm(33, , 77) =) = 2121

lcm(lcm(44, , 66) =) = 1212

lcm(lcm(55, , 1010) =) = 1010

Least Common Multiples

•Using prime factorizations:

•a = p1a

1 p2a

2 … pna

n , b = p1b

1 p2b

2 … pnb

n ,

•where p1 < p2 < … < pn and ai, bi ∈ N for 1 ≤ i ≤ n

•lcm(a, b) = p1max(a

1, b

1) p2

max(a2, b

2) … pn

max(an, b

n)

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•Example:

a = a = 60 60 = = 2222 3311 5511

b = b = 54 54 = = 2211 3333 5500

lcm(a, b) = lcm(a, b) = 2222 3333 551 1 = = 44**2727**5 5 = = 540540

GCD and LCM

a = a = 60 60 = = 222 2 3311 5511

b = b = 54 54 = = 2211 333 3 5500

gcd(a, b) = gcd(a, b) = 2211 3311 550 0 = = 66

22February 9, 2012Applied Discrete Mathematics

Week 3: Algorithms 22

lcm(a, b) = lcm(a, b) = 2222 3333 551 1 = = 540540

gcd(a, b) = gcd(a, b) = 22 33 55 = = 66

Theorem: Theorem: a*b a*b == gcdgcd((a,ba,b)* lcm()* lcm(a,ba,b))

Greatest Common Divisor

Relatively Prime

• Let a,b be integers, not both zero. The

greatest common divisor of a and b (or

gcd(a,b) ) is the biggest number d which divides

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gcd(a,b) ) is the biggest number d which divides

both a and b.

• Equivalently: gcd(a,b) is smallest number which

divisibly by any x dividing both a and b.

• a and b are said to be relatively prime if

gcd(a,b) = 1, so no prime common divisors.

Greatest Common Divisor

Relatively Prime

Q: Find the following gcd’s:

1. gcd(11,77)

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1. gcd(11,77)

2. gcd(33,77)

3. gcd(24,36)

4. gcd(24,25)

Greatest Common Divisor

Relatively PrimeA:

1. gcd(11,77) = 11

2. gcd(33,77) = 11

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3. gcd(24,36) = 12

4. gcd(24,25) = 1. Therefore 24 and 25 are relatively

prime.

NOTE: A prime number are relatively prime to all other

numbers which it doesn’t divide.

Greatest Common Divisor

Relatively Prime

Find gcd(98,420).

Find prime decomposition of each number and find all

the common factors:

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the common factors:

98 = 2·49 = 2·7·7

420 = 2·210 = 2·2·105 = 2·2·3·35

= 2·2·3·5·7

Underline common factors: 2·7·7, 2·2·3·5·7

Therefore, gcd(98,420) = 14

Greatest Common Divisor

Relatively Prime

Pairwise relatively prime: the numbers a, b, c, d, … are said to be pairwise relatively prime if

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d, … are said to be pairwise relatively prime if any two distinct numbers in the list are relatively prime.

Q: Find a maximal pairwise relatively prime subset of

{ 44, 28, 21, 15, 169, 17 }

Greatest Common Divisor

Relatively Prime

A: A maximal pairwise relatively prime subset of

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A: A maximal pairwise relatively prime subset of

{44, 28, 21, 15, 169, 17} :

{17, 169, 28, 15} is one answer.

{17, 169, 44, 15} is another answer.

Least Common Multiple

� The least common multiple of a, and b (lcm(a,b) )

is the smallest number m which is divisible by both

a and b.

� Equivalently: lcm(a,b) is biggest number which

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Equivalently: lcm(a,b) is biggest number which

divides any x divisible by both a and b

Q: Find the lcm’s:

1. lcm(10,100)

2. lcm(7,5)

3. lcm(9,21)

Least Common Multiple

A:

1. lcm(10,100) = 100

2. lcm(7,5) = 35

3. lcm(9,21) = 63

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3. lcm(9,21) = 63

THM: lcm(a,b) = ab / gcd(a,b)