Number theory:
for the first lecture of Number theory kindly read: Number Theory  Set of Natural NumbersĀ
As number theory is a concern of a set of natural numbers and we apply different formulas on numbers like wellordering principle.
The wellordering principle is derived from the induction method.
Similarly that of about DMAS rule. āDā stands for divides, āMā stands for multiplication, āAā stands for addition and āSā stands for subtraction.
DIVISIBILITY:
Number Theory involves Divisibility.
Let āaā and ābā be any two integers with āaā which is not equal to zero. Then āaā is said to divide ābā written (ab) if there is an integer ācā such that b=ac. If no such integer can be found when we say āaā does not divide ābā. āaā is called a divisor or a factor of ābā and ābā is called a multiple of āaā. Also if ab, then there is an integer c such that b=ac
Now if b=ac
Then b= (a)(c)
Then āab
Example:
The following examples illustrate the concept of divisibility of integers:
 13182
 530
 17289
 6 is not divided by 44
 7 is not divided by 50
 333
 170
Look at the last example part 7 which is 170, 0 is multiple and 17 is divisor.
Uses of Natural Numbers in Real Life:
 We use it to count anything.
 It is used for time measurement (minutes, seconds, hours, nanosecond, etc.)
 Used for counting living and nonliving things.
 Used for buying anything.
 Used as a time of arrival of the airplane.
 Position of anything just like Pakistan is on 3^{rd} position in population.
Examples:
The divisors of 6 are 1,2,3 and 6
The divisors of 17 are 1 and 17.
The divisors of 100 are 1, 2, 4, 5, 10, 20, 25, 50 and 100.
THEOREMS:

Reflexive property: Ā nn

Transitive property: Ā dn and nm then dm

Linearity property: dn and dm then d(an+bm)

Multiplication property: dn implies adan

Cancellation property: adan and a is not equal to zero which implies dn

1 divides every integer: 1n

Every integer divides zero: n0

Comparison property: dn and n is not equal to zero implies d<n

Equality: dn and nd implies d=n

Division conjugate: dn and nd then (nd)n
Proposition:
If a,b and c are integers with ab and bc, then ac.
Proof:
Since ab and bc by definition, there are integers e and f such that b=ae and c=bf
Hence bf=(ae)fĀ Ā Ā byĀ Ā b=ae;
And bf=(ef)
Then bf=bc(e(cb))
And then bf=c
And we conclude that ac
Example: 1166 and 66198Ā Ā thenĀ Ā 11198.
Proposition:
If a, b, m and n are integers and if caĀ Ā and cb then c(ma+nb)
Proof:
Since caĀ Ā and cb, then by definition there are integersĀ Ā Ā eĀ Ā andĀ Ā Ā fĀ Ā such thatĀ Ā a=ceĀ Ā Ā andĀ Ā b=cf
HenceĀ Ā ma+nbĀ =Ā Ā m(ce) + n(cf)
Consequently we see that c(ma+nb).
DIVISION OF ALGORITHM:
Number Theory involves the division of Algorithm too.
Let āaā and ābā be integers such that b is greater than zero. Then there exists a unique integerĀ Ā āqāĀ andĀ Ā ārā such that a=bq+r, where the range of r is from 0 to ābāĀ such that 0<r<b. Ā and moreover ārā is remainder and āqā is quotient.
GREATEST COMMON DIVISOR:
Let āaāĀ Ā and ābāĀ Ā be any two integers at least one of which is nonzero. Then their greatest common divisor Ā (GCD) is a positive integer ādā such that
 da and db
 If ca and cb then d>c that is if c is a common divisor of a and b then cd.
For further detail of greatest common divisor read this: Greatest Common Divisor GCD
COPRIME:
IfĀ āaāĀ and ābĀ are any two integers such that at least one of which is a nonzero then we say āaā and ābāĀ are coprime if their greatest common divisor is one. It is denoted as (a,b)=1 . coprime numbers are also called relative prime numbers.
COROLLARY:
If ac and bc and (a, b)= 1 then abc. but the conclusion will be false if āaā and ābā do not have 1 as a greatest common divisor.
PROOF:
Let ac and bc
Then there exist intergers ārā andĀ āsāĀ such that
We can say c= arĀ Ā andĀ Ā Ā Ā c= bs
Also (a, b)=1
This implies that there exist integersĀ āuā andĀ āvā such that
1= au+bv
This implies that c*1 = c a u + c b v
Then c= bs a u + ar b v
Then c + ab (s u + r v)
This implies abc
EXAMPLE:
We have a= 1 and b=4 and c=1
Then as we see (a, b)=1
As ac = 11
And bc = 41
Then abc= 4(1)1
EXAMPLE NUMBER 2:
We have a= 5 and b= 7 then c=35
Then as we see (a, b)=1
Then ac = 535
And bc = 735
Then abc = 5(7)35.
EXAMPLE NUMBER 3:
If we take a = 2, b= 4 and c= 12 then GCD of ab is not 1, it’s 2 so as ac and bc then ab does not divide c.
EUCLIDāS LEMMA:
If abc and (a, b)=1 then ac.
NOTE: Euclidās Lemma is false when GCD of a and b is not 1.
LEMMA:
Let d>1 be the smallest positive divisor of an integer ānā, then d is prime.
Proof:
LetĀ ādā ā by any divisor ofĀ ādā. then 1<dā<d
Suppose dā is not equal to 1 then dād and d n then by the property dān but also dā<d.
ButĀ ānā has the smallest divisor ādā. So,Ā dān and dā < d which implies dā=d
Hence the only divisor of d are 1 and d itself
Which implies ādā is prime.
NOTE:
Thus each integer >1 is either a prime or divisible by a prime.
EXAMPLE:
2 is a prime number having 2 divisors.
3 is a prime number having 1 divisor.
4 is not a prime number yet it has 3 divisors and that is 1, 2, 4 where 1 and 2 both are prime numbers.
THEOREM:
LetĀ āPā be a prime number andĀ Ā āaā any integer. Then either Pa or (a, P)=1
PROOF:
If P a, we have nothing to prove but if P does not divide āaāĀ Ā then we show that (a,Ā P) = 1.
LetĀ d=(a,Ā P)
Which implies da and dP
Since P is a prime number and d=1 or d=P
If d=P then P a, not possible
Hence d=1
So (a,Ā P)=1
COROLLARY:
If P is a prime and Pab then Pa or Pb.
PRIME DIVISIBILITY PROPERTY:
LetĀ āPāĀ be a prime number and suppose that P divides the product a1.a2.a3ā¦ar.Ā Then P divides at least one of the factor a1, a2, ā¦, ar
PROOF:
If āPā divides a1, we are done.
If not, we apply the claim to the product as a1(a2a3ā¦ar) to conclude that P must divide a2.a3.a4ā¦ar.
In other words, we are applying the claim with a=a1 and b= a2a3ā¦ar, as we know that Pabs p, if P does not, divides āaā then P must divide b as Pb.
So, now we know that P divides a2a3a4ā¦ar. If P divides a2 then we are done but if P does not divide a2 then P must divide a3a4a5ā¦ar as a=a2 and b=a3a4a5ā¦ar and this will continue till we find any number which is divided by any number in the product so that we will eventually find a number that is divided by P.
Number theory is the study of the set of the Natural Numbers and we can apply many operation and rules on numbers.
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