introduction to number theory - wordpress.com · introduction to number theory •number theory is...
TRANSCRIPT
![Page 1: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/1.jpg)
Introduction to Number Theory
•Number theory is about integers and their properties.
•We will start with the basic principles of
2 2
•We will start with the basic principles of
• divisibility,
• greatest common divisors,
• least common multiples, and
• modular arithmetic
![Page 2: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/2.jpg)
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
3 3
•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.
![Page 3: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/3.jpg)
Divisors.
ExamplesQ: Which of the following is true?
1. 77 | 7
2. 7 | 77
3. 24 | 24
4L9 4
3. 24 | 24
4. 0 | 24
5. 24 | 0
![Page 4: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/4.jpg)
Divisors.
ExamplesA:
1. 77 | 7: false bigger number can’t divide smaller positive number
2. 7 | 77: true because 77 = 7 · 11
5L9 5
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)
![Page 5: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/5.jpg)
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.
6 6
• 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.
![Page 6: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/6.jpg)
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
7L9 7
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
![Page 7: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/7.jpg)
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
8 8
•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.
![Page 8: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/8.jpg)
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
![Page 9: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/9.jpg)
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
![Page 10: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/10.jpg)
Division
Remember long division?
3
11731
q the quotient
d the divisor
11L9 11
117 = 31·3 + 24
a = dq + r
11731
24
93r the
remainder
a the dividend
![Page 11: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/11.jpg)
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
12L9 12
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.
![Page 12: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/12.jpg)
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,
13 13
• d is called the divisor,
• a is called the dividend,
• q is called the quotient, and
• r is called the remainder.
![Page 13: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/13.jpg)
The Division Algorithm
•Example:
•When we divide 17 by 5, we have
14
•17 = 5⋅3 + 2.
• 17 is the dividend,
• 5 is the divisor,
• 3 is called the quotient, and
• 2 is called the remainder.
![Page 14: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/14.jpg)
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.
![Page 15: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/15.jpg)
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.
![Page 16: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/16.jpg)
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)
17 17
•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
![Page 17: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/17.jpg)
Relatively Prime Integers•Definition:
•Two integers a and b are relatively prime if
gcd(a, b) = 1.
•Examples:
•Are 15 and 28 relatively prime?
18 18
•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.
![Page 18: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/18.jpg)
Relatively Prime Integers
•Definition:
•The integers a1, a2, …, an are pairwise relatively prime if
gcd(ai, aj) = 1 whenever 1 ≤ i < j ≤ n.
•Examples:
19 19
•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.
![Page 19: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/19.jpg)
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
![Page 20: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/20.jpg)
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)
21February 9, 2012 21
•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
![Page 21: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/21.jpg)
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))
![Page 22: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/22.jpg)
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
23L9 23
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.
![Page 23: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/23.jpg)
Greatest Common Divisor
Relatively Prime
Q: Find the following gcd’s:
1. gcd(11,77)
24L9 24
1. gcd(11,77)
2. gcd(33,77)
3. gcd(24,36)
4. gcd(24,25)
![Page 24: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/24.jpg)
Greatest Common Divisor
Relatively PrimeA:
1. gcd(11,77) = 11
2. gcd(33,77) = 11
25L9 25
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.
![Page 25: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/25.jpg)
Greatest Common Divisor
Relatively Prime
Find gcd(98,420).
Find prime decomposition of each number and find all
the common factors:
26L9 26
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
![Page 26: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/26.jpg)
Greatest Common Divisor
Relatively Prime
Pairwise relatively prime: the numbers a, b, c, d, … are said to be pairwise relatively prime if
27L9 27
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 }
![Page 27: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/27.jpg)
Greatest Common Divisor
Relatively Prime
A: A maximal pairwise relatively prime subset of
28L9 28
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.
![Page 28: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/28.jpg)
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
29L9 29
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)
![Page 29: Introduction to Number Theory - WordPress.com · Introduction to Number Theory •Number theory is about integers and their properties. •We will start with the basic principles](https://reader034.vdocuments.us/reader034/viewer/2022042503/5f7d37ea1832c3027a3043b7/html5/thumbnails/29.jpg)
Least Common Multiple
A:
1. lcm(10,100) = 100
2. lcm(7,5) = 35
3. lcm(9,21) = 63
30L9 30
3. lcm(9,21) = 63
THM: lcm(a,b) = ab / gcd(a,b)