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.

SOCIAL ARITHMETIC

A commerce activity in economy will connect directly with the rule and calculation whose the definition and the solution need mathematic. For example, a manager of self-service store usually takes 30% of profit. If one day those store obtain 20 million rupiahs of result selling, how much the profit of this store in that day?

• Prerequisite Matter

Before you learn about social arithmetic in this chapter, it is necessary to review about count operation of integer number, fragment number, equation and algebra operation which have learned. Those matter is a basic to learn the matter in this chapter. So, try to solve this problem bellow!

1. Do the addition and substraction operation of this number!
   
   a. 4500 + 7500 c. 25.000 – 16.500
   b. 24,36 + 18,29 d. 45,24 – 22,48

2. Do the multiply and division operation of this number!
   a. 15 x 7500 c. 12.500 : 200

   b. 2,5 + 40.000 d. 5.700.000 : 300

3. Define the result of multiply bellow!
   a. 1/5+ 40.000 c. 0,12 – 60.000

   b. 12(1/2)x 80.000 d. 0,09 – 500.000

4. Change this number into percent type!
   a. 3/5 c.2/15

   b. 7/40 d.11/30

5. Change this number until the nearest unit!
   a. 16,73 c. 2,7468
   b. 12,46 d. 2,9996

6. Change this number into ordinary fragment!
   a. 20% c. 7(1/2)%
   b. 50% d. 16(2/3)%

7. Do this calculation!
   a. 5% of 50.000 c.12(1/2) % of 400.000

   b. 20% of 25.000 d. 33(1/2)% of 2.000.000

8. Simplify this algebra type!
   a. 7x + 8x c. 4(3p - 2) + 5p
   b. 12x – 4x + 8x d. 8 – 3(2p + 4)

9. Do this equations!
   a. 6x + 5 = 13 c. 9y – 7y = 48
   b. 8x – 7 = 21 d. 10y + 2y = 20

If you haven’t competenced the matter above, please you learn again some of problems which resemble, in order to you can competence this this social arithmetic lesson easier.

• Buying Cost, Selling Cost, Profit, and Loss

In commerce activity there are goods seller and the buyer. The seller gives the goods to the buyer, then the buyer gives money as chance the goods which has been accepted.

To get the goods which will be sell, the seller buy from factory, distributor, or another places is called as buying cost or capital. Then, the money which accepted by the seller as the result of selling of this goods called selling cost. So, a commerce activity always has connection with buying cost or capital which will be the basic calculation.

In a commerce, there are two possibilities which will happen to the seller, there are

1. the seller will get profit, or

2. the seller will suffer a financial loss

• Profit

To understand about definition of profit, observe the explanation bellow!

A school cooperation buys a box of mineral water which contain 48 glasses with 14.000 rupiahs in cost. Its then be sold 500 rupiahs in cost per glass. Compare the buying cost and the selling cost!

The buying cost = Rp 14.000,-

The selling cost = 48 x Rp 500 = Rp 24.000,-

Obviously, the selling cost is higher than buying cost.

Difference between selling and buying cost = Rp 24.000 – Rp 14.000

= Rp 10.000

In this case, the school cooperation get profit Rp 10.000,-. Based on the explanation above, we may conclude that the seller get a profit when the selling cost is higher than the buying cost.

Profit = selling cost – buying cost

Example :

1. A dozen of pencil is bought Rp 18.000 in cost, then its sold Rp 1.800 in cost for each pencil. How much the profit?

Problem solving :

Given the buying cost = Rp 18.000,-

Selling cost = 12 x Rp 1.800 =Rp 21.600

Profit = selling cost – buying cost

= Rp 21.600 – Rp 18.000 = Rp 3.600

2. A seller buy two kinds of rice 65 kg for each sort, it’s cost Rp 3.800 @ kg and the other 35 kg, it’s Rp 4.000 @ kg. both of the rices then are mixed each other and be sold Rp 4.200 @ kg. how much the profit which be reached by the seller?

Problem solving :

buying cost = (65 x Rp 3.800) + (35 x Rp 4.000)

= Rp 247.000 – Rp 140.000

= Rp 387.000

Selling cost = (65 + 35) x Rp 4.200

= 100 x Rp 4.200

= Rp 420.000

Profit = Rp 420.000 - Rp 387.000 = Rp 33.000

• Loss

Mr. Mamat buy a secondhand television Rp 250.000 in cost. It’s repaired and spent Rp 65.000. then it’s sold Rp 300.000 in cost. If the repairing cost and buying cost calculate as capital, so :

Capital of that television = Rp 250.000 - Rp 65.000

= Rp 315.000

Selling cost = Rp 300.000

The selling cost is lower than the capital, and in this case is called that Mr. Mamat suffer a financial loss.

Difference between selling and buying cost = Rp 315.000 – Rp 300.000

= Rp 15.000

So, Mr. Mamat has a financial loss Rp 15.000. Based on the case above, we may conclude that the seller suffer a financial loss if the selling cost is lower than buying cost or capital.

Loss = buying cost – selling cost

• • Buying Cost and Selling Cost

In the profit and loss chapter have been explained that the profit and loss are calculable if selling cost and buying cost are known.

In the commerce, the profit of selling cost can be obtained if the selling cost is higher than buying cost. Because the selling cost is higher than buying cost, and the profit as great as dispute between selling cost and buying cost, then may be obtained this relationship :

selling cost = buying cost + profit

o

buying cost = selling cost - loss

r

next, if the commerce suffer a loss, then selling cost is lower than buying cost, and the loss is the dispute between buying cost and selling cost, so we can obtain this relationship.

selling cost = buying cost - loss