المحاضرة السابعة هياكل متقطعة

Number theory is about integers and their properties.

We will start with the basic principles of

divisibility,
greatest common divisors,
least common multiples, and
modular arithmetic

and look at some relevant algorithms.
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 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
(see book for correct symbol).
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.

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.


Number theory is about integers and their properties.

We will start with the basic principles of

divisibility,
greatest common divisors,
least common multiples, and
modular arithmetic

and look at some relevant algorithms.
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 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
(see book for correct symbol).
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.

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.


Number theory is about integers and their properties.

We will start with the basic principles of

divisibility,
greatest common divisors,
least common multiples, and
modular arithmetic

and look at some relevant algorithms.
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 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
(see book for correct symbol).
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.

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.
Number theory is about integers and their properties.

We will start with the basic principles of

divisibility,
greatest common divisors,
least common multiples, and
modular arithmetic

and look at some relevant algorithms.
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 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
(see book for correct symbol).
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.

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.


Number theory is about integers and their properties.

We will start with the basic principles of

divisibility,
greatest common divisors,
least common multiples, and
modular arithmetic

and look at some relevant algorithms.
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 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
(see book for correct symbol).
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.

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.


Number theory is about integers and their properties.

We will start with the basic principles of

divisibility,
greatest common divisors,
least common multiples, and
modular arithmetic

and look at some relevant algorithms.
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 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
(see book for correct symbol).
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.

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.


Number theory is about integers and their properties.

We will start with the basic principles of

divisibility,
greatest common divisors,
least common multiples, and
modular arithmetic

and look at some relevant algorithms.
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 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
(see book for correct symbol).
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.

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.


Number theory is about integers and their properties.

We will start with the basic principles of

divisibility,
greatest common divisors,
least common multiples, and
modular arithmetic

and look at some relevant algorithms.
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 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
(see book for correct symbol).
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.

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.