FACTORIZATION NUMBER OF INTEGER


• Prime Number

We have known that positive number of integer which prime one another, that is the greatest alliance factor from that both number is equal 1.
If is positive number of integer so that ( ) = 1, then it’s may be called that prime one another too. But, if ( ) = 1, for each i,j = 1, 2, 3…, n with , so it may be called that the positive numbers of integer are prime one another pair to pair.

Example 4.1

1) because (5, 8, 9) = 1, so 5, 8, and 9 are called as three numbers which prime each another along with prime one another pair to pair, because (5, 8) = (5, 9) = (8, 9) = 1  
 
2) because (3, 9, 4, 8) = 1, so 3, 4, 8, and 9 are called as four numbers which prime one another, but not one another pair to pair, because (3, 9) = 3 and (4, 8) = 4, although (3, 4) = (3, 8) = (9, 4) = (9, 8) = 1

let a and b are positive number of integer , so based on division of algorithm, there are integer numbers of q and r so that

q = qa + r with 0≤ r <>

known that (a, r) = 1, so based on theorem 2.8, we can conclude that (a, b) =1.it may be called that if the residue of division b by a is prime to a, so b is also prime to a. 

Definition 4.1 :
The positive number of integer which more than 1 and do not have positive factor of integer except 1 and itself number of integer is called prime number. Positive number of integer which more than 1 and not as prime number is called as composite number.

The series of prime numbers : 2, 3, 5, 7, 11, 13, 17, …
The series of composite numbers : 4, 6, 8, 10, 12, 14, 15, …

Pay attention that 1 is not a prime number and also not a composite number. 1 is called as unit. So, the set of all positive number of integer (original number) divided into set of part which independent one another, that is 
1) Set of whole prime number
2) Set of whole composite number
3) Set of unit

Pay attention to a positive number of integer, for example 210, then 210 is apartable into prime factors, that is :
210 = 2. 3. 5. 7 or
210 = 3. 7. 2. 5 or
210 = 7. 3. 5. 2 and another one.
The difference of apartion of 210 into its prime factors is just in arrangement of its factors. It is an example that a positive number which more than 1 is may be sign as multiplication of fixed prime number. The multiplication form of prime number is singular, except the arrangement of that prime numbers. It’s often called by singular factorization theorem.