Mathematics Number Theory’s Theorem

a ’ T e e - If P is positive prime and a is any integer such that P is not a ,

divisor of a do that

≡

.

= , therefore if the no. . . ………… − , are divided by P. The remainders are . . … … … … − , placed in some order so that.

Prove: Since The product

Now

.

,

−

= , then

…………

−

, the product . .

…………

−

……………

≡

≡

−−−−−−−−

−

−−−−−−− ≡

Multiplying these congruence relations,

�

. . …………

.

−

−

.

………… −

!

Since P is prime, therefore,

−

=

≡

−

=

.

. . ………… −

And Hence

By cancelling P −

−

,

!

, ,

,

−

−

−

= ,

…………

�

−

�

……….

……….

= ,

= ,

! =

! from both sides of the congruence relation (2), −

|

Lokesh Kumar (Computer Science Part 1) Roll no. 17

≡

−

−

=

−

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Mathematics Number Theory’s Theorem e ’s Theorem- If m is a positive integer and a is any integer such that ,

then �

=

.

Proof: Let { .

……….

and consequently each

�

= ,

} be a reduced set of residues set of residues modulo m

is congruent modulo m to one and only one

.

Let ≡

,

≡

,

−−−−−−−−− −−−−−−−−− .

…………

−−−−−−−−−

…………

�

, are precisely

�

.

so that, the product

.

≡

.

………

.

= ��� ���d�c� ��

.

………… ……… � …………

placed in some order

� �

M���������� ��� ����� c�������c��.

………

�

≡

�

.

.

�

�

≡

.

≡

.

………

…………

.

,

�

�

………

( = . …………� ) is relativity prime to m, therefore their product . ………… � is also relatively prime to m, so cancelling . ………… � from both sides of the congruence relation (2).

Since each

W

�

=

−

!+

’ T e e - If p is a positive prime, then = , then

Case 1: If

Case 2: Let

⇒ . . −

!=

⇒

−

�� � �������� �� �.

and so the statement

≡

�� ����.

> , Let a be a positive integer such that

prime, therefore a and P are relatively prime

Since

,

modulo P

⇒

,

=

= , therefore the linear congruence

Lokesh Kumar (Computer Science Part 1) Roll no. 17

!+

≡

≡

.

+ . − , Since P is a , has a unique solution

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Mathematics Number Theory’s Theorem

⇒ ����� ������ � ������ ������� ��� �, ��c� ���� ��d

���� � ≠

−

≡

��c���� � d��� ��� ������� ≡

These integers a and b are inverse modulo P of each others.

=

Suppose that

⇒ ������� ��d��� P �� � �� � ������. T��� ⇔

I��

=

⇒

=

����

⇒

I��

S��c�

And

≠

≡

��� �

|

��d

−

− |

−

+

−

[ P �� � �����] �� ��������

|

|

−

≠

��d

− ,

+

⇒

=

−

�� ��������

�. �. ������� ��d��� P �� � �� � ������

B�c���� ������� ��d��� Such that

+

|

|

⇔

|

I��

≠

⇔

≡

I�

=

��d ���� ��

��

��d

≠

=

−

c�� ��� �� −

���

, �� ��� ��� �� � = { , , … . . ,

−

��d

����� ������ � ������ ��. −

The P-3 elements in the set S from

��

− }

−

−

�����������.

pairs of distinct elements of S.

such that the product of each pair is congruent modulo P to 1. Multiplying the

−

congruencies,

⇒ . ,……………

. . …………

−

−

−

!≡

!+

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− ≡

−

−

≡

≡

−

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Mathematics Number Theory’s Theorem ��

La a

− [

�� � d����� ��

!+ ≡

−

e’ Theorem-

≡

!+ ,

−

Statement- If P is a positive prime integer< the no. , , , … … … ,

=

Proof- Let

Where �

−

=

+

]

!+

− , then the sum of the products of

− , yaken a at a time, is divisible by P. +

−

+

+�

−

replacing

by

+

=

+ ,

+�

+ ̅̅̅̅̅̅̅̅ −

………………

+. … … … … + �

+�

−

………………

……………

−

is the sum of the products of the numbers , , , … … … ,

From . . . . . . . . . .

+

+

=

+

+

+

+

+

=

+

+

+

[

−

+ =

+� + [

−

equating the coefficient of

⇒

−

,

+� .

+� .

−

+ +� −

−

−

−

+� .

−

+�

+ +�

−

+

+� − ] ] …………

on both sides of the identity 4.

+� ≡ � +�

+� .

……………

+. … … … … … + � − − +. … … … … + � −

,………….

− , taken r times.

+ ̅̅̅̅̅̅̅̅ − . … … … … + ̅̅̅̅̅̅̅̅ −

…………

Substituting for f(x) and f(x+1) from (2) in (3)

+

�� � �������� �� .

−

+�

−

≡ �

………

………

≡ �

−−−−−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−−−−−

A����

N��

−

| � . |

−

⇒ |� , S��c�

| � . |

⇒

. |� . −

⇒ |� , B�c����

−

��d

S�������� �� c�� ���� ���� I�

= , , ,…………,

+�

−

|� . −

−

.

≡

=

�

−

�����

⇒ |� . −

��� ���������� P���� |� , |� , … … … … |� − .

− , �ℎ��

d���d�� � , �����,

� d����� ��� ��� �� ��� ���d�c�� �� ��� ��. , , , … … T���� � �� � �����.

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����� − ,

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Mathematics Number Theory’s Theorem

Definition of Devisors- Let a, b be two integer and ≠ . It there exists an integer c such that

=

, then we say.

That a divides b or a is a divisor of b or a is a factor of b is a multiple of a. When a is a divisor of b, we write ˶ | ˷.

This is read as a is a divisor of b, then b is a multiple of a and its write as

For example1. 2|8 since 8=2.4 when 2. -4|16 since 3.

×

= −

∈ .

. −

Theorem of Division Algorithm-

Proof- Let the set = { − =

− .

Therefore at least If

− .| |

If i.e.

If i.e.

: ∈ }. ∈

<

∈ .

and thus s is not empty.

. .,

. . − .| | ∈ <

Now

.

< | |.

+ .

.| |

−

then

−| | − .| |

| |∈

< , then s contains at least one non negative integer.

If

=

≠ , then there exists unique integer q.r such that. =

Since

.

.

is 3 is not divisor of 4 because there exists no integer q such that

If a is any integer and

=

− .| |

i.e. , if

, then

−| |

−| | − . −| |

− . −| | ∈

> , then s contain at least one non-negative integer. − . −| |

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.

−| |∈

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Mathematics Number Theory’s Theorem >

<

The set always contains non-negative integers. The principle well-ordering, the non-empty subset of consisting of non-negative integers has a least member. Let negative integer belonging to s.

=

−

∈ , be the smallest non-

Since r is non-negative.

,

−| |∈ ,

Show that

> ,

If

=

−

>

−

,

If

−| |=

−

∈ ,

≠

−| |<

< , −| |

−| |

i.e.

<

| |,

<| | −

−

−

∈ .

.

− | | is non-negative integer belonging to s and − | | < .

R as he smallest non-negative integer ∈ . We have

<| |

Integers q and r such that

=

. .

−

=

+ <| |

Now to show that the integers q and r are unique. Let the another pair Such that Then

′ and ′.

=

′

+ ′

+

′

′

,

′

′

=

+

<| |

− = − ′ | − ′ |

Then Lokesh Kumar (Computer Science Part 1) Roll no. 17

′

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Mathematics Number Theory’s Theorem <| |

′

Now putting We get ′

=

′

=

− ′| |

− ′ is possible only if

r is a divisor of

=

′

in

′

−

′

′

=

′

+

=

=

by applying cancellation law.

≠

Since ′

and

=

′<| |

and there

=

+

and

are

unique.

Positive primes in at least one no way: Uniqueness of prime factor: Now we prove that the factorisation is unique. It possible let a second factorisation of a as a product of positive prime be given by. ′

Then

′

………… ′

⟹

′

|

′

=

=

′

|(

′

′ ′

…………

…………

…………

…………

′ ′

′

′

Since

′

and

′

Then ′

We get

/

|

are both positive primes. ′

⟹

…………

.

Since the product of integers is commutative. Let

)|

…………

′

…………

′

= +

…………

′

=

′

=

′

⟹

=

…………

…………

…………

,

…………

And also the equality of a prime factor in each product to some factor in the other product. A unique factorisation of a except for the order of the prime factor. If

=

And every

……… =

=

…………

for some j and every

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are two prime factorisations for a

=

=

for some i.

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Mathematics Number Theory’s Theorem

Proved

The fundamental theorem of Arithmetic or the unique factorisation theorem: >

Statement: Every positive integer positive primes.

can be expressed uniquely as a product of

Proof: Existence of prime factorisation: First we shall prove that a can be factored at least in one way as the product of positive primes, by induction on a. The least positive integers > If

is 2.

=

Since 2 itself prime. Now assume according to induction hypothesis that the theorem is true for all integers k. Such that

<

<

And let integer a. If a is prime. Then if a is not prime. So Where

=

<

<

<

and

<

According to induction hypothesis b and c can be expressed as products of positive primes. Let

=

Where Then

=

′

=

…………… and

′

and

=

……………

are positive prime.

…………

…………

is a product of positive prime.

Such that by mathematical induction every positive integers a greater than 1 can be expressed as a product of.

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Mathematics Number Theory’s Theorem

Great common divisor: Let a and b be two integers not both zero i.e. at least one of them is not equal to zero. Then the greatest common divisor (G.C.D.) of a and b is a positive integer d, such that, 1. d|a and d|b i.e. d is a common divisor of a & b and, 2. If, for an integer c, c|a and c|b, then c|d i.e. every common divisor of a and b is a divisor of d. If d is the G.C.D. of a and b, then symbolically we write, Thus

,

=

,

will be read as G.C.D. of a and b. The greatest common divisor is

sometimes also called the highest common factor (H.C.F.). Example: 1. 1, 2, 3, 4, 6 and 12 are common divisors of 24 & 60. 2. Out of these each of 1, 2, 3, 4 and 6 is a divisor of 12.

Euclidean Algorithm: Euclidean algorithm enables us to find the actual value of the greatest common divisor d of two given integers a and b, also to find integers x and y. Such that

=

+

Let a, b be two integers such that at least one of a, b is non-zero.

= , then

If

Case 1:

If Case 2: If Since

,

≠

= , then

and

= | |, | |

≠ ,

,

,

=| |

=| |

We can assume that both a and b are positive integers. Using division algorithm, finite chain of divisions. Lokesh Kumar (Computer Science Part 1) Roll no. 17

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Mathematics Number Theory’s Theorem =

=

+ ,

+

=

,<

,

+

,<

,

,<

………………………………………………………… ………………………………………………………… −

−

=

−

=

>

+

>

+

+

+

>

,

,

,< +

>

Some r must be zero.

+

This process must terminates in the

⟹

⟹

⟹

,<

………………

and only finitely many integers lie between 0 & b.

⟹

−

process.

−

= =

=

−

−

+

.

Where

step.

, is the last non-zero remainder in repeated division

Let know that

,

=

So that last non-zero remainder

=

,

=. … … … … … =

−

,

=

is the G.C.D. of the given integer a and b.

Q. Find (26,118) and express it in the from 26x+118y, where x and y Ans. Using the process of division algorithm.

=

. +

= So the last non-zero remainder

=

=

Put the value of 12 is eg (5)

=

−

−

−

…………………

. + …………………

=

−

∈ .

…………………

. +

. + ………………… =

From eq (1) & (2).

=

.

.

.

=

=

= .

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,

.

−

−

………………

………………

−

………………

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Mathematics Number Theory’s Theorem

From eq (1)

Put his value of 14 in eq (7)

= [ And if

=

=

= [

+

,

⟹

−

.

. ]− ] ]−[

−

=

=− ,

=

Ans.

Theorem: Two integers a and b are relatively prime if and only if we can find integers x

+

and y such that

= .

,

Proof: Let a and b are two relatively prime, then Let

⟹

,

=

=

,

⟹

| , |

Since the G.C.D. is to be positive, Hence

,

=

⟹

⟹

⟹

d=1

|

= .

+

|

=±

and so a & b are relatively prime.

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