OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

Introduction to Number Theory I

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

Divisibility

Greatest common divisor

Euclidean algorithm

Extended Euclidean algorithm

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

DefinitionLet a and b be integers. We say that a 6= 0 divides b if there is aninteger k such that a · k = b. The notation for this is a|b.

Example

7|63 because 7 · 9 = 63.

Note that every number divides zero since a · 0 = 0 for everyinteger a.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

LemmaFor integers a, b, c, the following hold:

I If a|b, then a|b · c for all c.

I If a|b and b|c, then a|c.

I If a|b and a|c, then a|(sb + tc) for all s and t.

I If a|b and b|a, then a = ±b.

I For all c 6= 0, a|b if and only if ca|cb.

Proof: Exercise.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

LemmaFor integers a, b, c, the following hold:

I If a|b, then a|b · c for all c.

I If a|b and b|c, then a|c.

I If a|b and a|c, then a|(sb + tc) for all s and t.

I If a|b and b|a, then a = ±b.

I For all c 6= 0, a|b if and only if ca|cb.

Proof: Exercise.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

LemmaFor integers a, b, c, the following hold:

I If a|b, then a|b · c for all c.

I If a|b and b|c, then a|c.

I If a|b and a|c, then a|(sb + tc) for all s and t.

I If a|b and b|a, then a = ±b.

I For all c 6= 0, a|b if and only if ca|cb.

Proof: Exercise.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

LemmaFor integers a, b, c, the following hold:

I If a|b, then a|b · c for all c.

I If a|b and b|c, then a|c.

I If a|b and a|c, then a|(sb + tc) for all s and t.

I If a|b and b|a, then a = ±b.

I For all c 6= 0, a|b if and only if ca|cb.

Proof: Exercise.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

LemmaFor integers a, b, c, the following hold:

I If a|b, then a|b · c for all c.

I If a|b and b|c, then a|c.

I If a|b and a|c, then a|(sb + tc) for all s and t.

I If a|b and b|a, then a = ±b.

I For all c 6= 0, a|b if and only if ca|cb.

Proof: Exercise.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

DefinitionA number p > 1 with no positive divisors other than itself and 1 iscalled a prime number.

Non-prime numbers bigger than 1 are called composite numbers.

The Prime Number theorem:

For each positive integer n there are approximately n/ ln n primenumbers up to n.

Euclid’s theorem:

If p is a prime number and p divides a · b, then p divides a or pdivides b.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

DefinitionA number p > 1 with no positive divisors other than itself and 1 iscalled a prime number.

Non-prime numbers bigger than 1 are called composite numbers.

The Prime Number theorem:

For each positive integer n there are approximately n/ ln n primenumbers up to n.

Euclid’s theorem:

If p is a prime number and p divides a · b, then p divides a or pdivides b.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

The Fundamental Theorem of Arithmetic:

Every positive integer n can be written in a unique way in the form

n = pe11 · pe2

2 · pe33 ....pek

k

where p1 ≤ p2 ≤ p3 ≤ ... ≤ pk are primes and e1, e2, e3, ..., ek arepositive integers.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

The Fundamental Theorem of Arithmetic:

Every positive integer n can be written in a unique way in the form

n = pe11 · pe2

2 · pe33 ....pek

k

where p1 ≤ p2 ≤ p3 ≤ ... ≤ pk are primes and e1, e2, e3, ..., ek arepositive integers.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

The Fundamental Theorem of Arithmetic:

Every positive integer n can be written in a unique way in the form

n = pe11 · pe2

2 · pe33 ....pek

k

where p1 ≤ p2 ≤ p3 ≤ ... ≤ pk are primes and e1, e2, e3, ..., ek arepositive integers.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

DefinitionA positive integer d is called greatest common divisor of a andb if d divides both a and b and any divisor of a and b is also adivisor of d . It is denoted gcd(a, b).

DefinitionIf gcd(a, b) = 1, then we say that a and b are relatively prime.

Dirichlet’s theorem:

Let a, b ∈ N be relatively prime. Then there exist infinitely manyprimes of the form a · n + b for n ∈ N.

Example

gcd(1539, 7380) = 32 since 1539 = 34 · 19 and 7380 = 22 · 32 · 5 · 41

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

DefinitionA positive integer d is called greatest common divisor of a andb if d divides both a and b and any divisor of a and b is also adivisor of d . It is denoted gcd(a, b).

DefinitionIf gcd(a, b) = 1, then we say that a and b are relatively prime.

Dirichlet’s theorem:

Let a, b ∈ N be relatively prime. Then there exist infinitely manyprimes of the form a · n + b for n ∈ N.

Example

gcd(1539, 7380) = 32 since 1539 = 34 · 19 and 7380 = 22 · 32 · 5 · 41

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

DefinitionA positive integer d is called greatest common divisor of a andb if d divides both a and b and any divisor of a and b is also adivisor of d . It is denoted gcd(a, b).

DefinitionIf gcd(a, b) = 1, then we say that a and b are relatively prime.

Dirichlet’s theorem:

Let a, b ∈ N be relatively prime. Then there exist infinitely manyprimes of the form a · n + b for n ∈ N.

Example

gcd(1539, 7380) = 32 since 1539 = 34 · 19 and 7380 = 22 · 32 · 5 · 41

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

LemmaIf a = q · b + r then gcd(a, b) = gcd(b, r).

Theorem (Euclid, 300 B.C.)

Given integers a and b with b > 0, there exist unique integers qand r satisfying

a = q · b + r , 0 ≤ r < b

The integers q and r are called, respectively, the quotient andremainder in the division of a and b.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

LemmaIf a = q · b + r then gcd(a, b) = gcd(b, r).

Theorem (Euclid, 300 B.C.)

Given integers a and b with b > 0, there exist unique integers qand r satisfying

a = q · b + r , 0 ≤ r < b

The integers q and r are called, respectively, the quotient andremainder in the division of a and b.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

Example

gcd(1147, 899) =

gcd(899, 248︸︷︷︸1147=1·899+248

)

= gcd(248, 155︸︷︷︸899=3·248+155

)

= gcd(155, 93︸︷︷︸248=1·155+93

)

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

Example

gcd(1147, 899) = gcd(899, 248︸︷︷︸1147=1·899+248

)

=

gcd(248, 155︸︷︷︸899=3·248+155

)

= gcd(155, 93︸︷︷︸248=1·155+93

)

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

Example

gcd(1147, 899) = gcd(899, 248︸︷︷︸1147=1·899+248

)

= gcd(248, 155︸︷︷︸899=3·248+155

)

=

gcd(155, 93︸︷︷︸248=1·155+93

)

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

Example

gcd(1147, 899) = gcd(899, 248︸︷︷︸1147=1·899+248

)

= gcd(248, 155︸︷︷︸899=3·248+155

)

= gcd(155, 93︸︷︷︸248=1·155+93

)

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

Example (Cont.)

= gcd(93, 62︸︷︷︸155=1·93+62

)

=

gcd(62, 31︸︷︷︸93=1·62+31

)

= gcd(31, 0︸︷︷︸62=2·31+0

)

= gcd(31, 0)

= 31.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

Example (Cont.)

= gcd(93, 62︸︷︷︸155=1·93+62

)

= gcd(62, 31︸︷︷︸93=1·62+31

)

=

gcd(31, 0︸︷︷︸62=2·31+0

)

= gcd(31, 0)

= 31.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

Example (Cont.)

= gcd(93, 62︸︷︷︸155=1·93+62

)

= gcd(62, 31︸︷︷︸93=1·62+31

)

= gcd(31, 0︸︷︷︸62=2·31+0

)

=

gcd(31, 0)

= 31.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

Example (Cont.)

= gcd(93, 62︸︷︷︸155=1·93+62

)

= gcd(62, 31︸︷︷︸93=1·62+31

)

= gcd(31, 0︸︷︷︸62=2·31+0

)

= gcd(31, 0)

= 31.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

TheoremLet a and b be positive integers with a ≥ b. The followingalgorithm computes gcd(a, b) in a finite number of steps.

1. Let r0 = a and r1 = b.

2. Set i=1.

3. Divide ri−1 by ri to get a quotient qi and remainder ri+1,

ri−1 = ri · qi + ri+1 with 0 ≤ ri+1 < ri

4. If the remainder ri+1 = 0, then ri = gcd(a, b) and thealgorithm terminates.

5. Otherwise, ri+1 > 0, so set i = i + 1 and go to Step 3.

The division step is executed in at most 2log2(b) + 1 times.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

TheoremLet a and b be positive integers with a ≥ b. The followingalgorithm computes gcd(a, b) in a finite number of steps.

1. Let r0 = a and r1 = b.

2. Set i=1.

3. Divide ri−1 by ri to get a quotient qi and remainder ri+1,

ri−1 = ri · qi + ri+1 with 0 ≤ ri+1 < ri

4. If the remainder ri+1 = 0, then ri = gcd(a, b) and thealgorithm terminates.

5. Otherwise, ri+1 > 0, so set i = i + 1 and go to Step 3.

The division step is executed in at most 2log2(b) + 1 times.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

TheoremLet a and b be positive integers with a ≥ b. The followingalgorithm computes gcd(a, b) in a finite number of steps.

1. Let r0 = a and r1 = b.

2. Set i=1.

3. Divide ri−1 by ri to get a quotient qi and remainder ri+1,

ri−1 = ri · qi + ri+1 with 0 ≤ ri+1 < ri

4. If the remainder ri+1 = 0, then ri = gcd(a, b) and thealgorithm terminates.

5. Otherwise, ri+1 > 0, so set i = i + 1 and go to Step 3.

The division step is executed in at most 2log2(b) + 1 times.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

LemmaThe following statements about the greatest common divisor hold:

I Every common divisor of a and b divides gcd(a, b).

I gcd(k · a, k · b) = k · gcd(a, b) for all k > 0.

I If gcd(a, b) = 1 and gcd(a, c) = 1,then gcd(a, b · c) = 1.

I If a|bc and gcd(a, b) = 1, then a|c.

Proof: Exercise.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

TheoremGiven integers a and b not both of which are zero, there existintegers x and y such that gcd(a, b) = a · x + b · y.

TheoremLet a and b be integers with at least one of them different fromzero. The integers a and b are relatively prime if and only if1 = a · x + b · y for some x , y ∈ Z.

Introduction to Number Theory I

OutlineDivisibility

Greatest common divisorEuclidean algorithm

Extended Euclidean algorithm

Example

gcd(12378, 3054) = 4 = 3186 · 12378 + (−12913) · 3054

TheoremLet a and b be positive integers. Then the equationax + by = gcd(a, b) always has a solution in integers x and y. If(x0, y0) is one solution, then every solution has the formx = x0 + b·k

gcd(a,b) and y = y0 − a·kgcd(a,b) for some k ∈ Z.

Proof: Exercise.

Introduction to Number Theory I