PRIME FACTORS AND PRIME FACTORIZATION

PRIME FACTORS AND PRIME FACTORIZATION

Recall that a prime number is a number greater than 1 and having no factors except 1 and itself

Prime Factors

The factors of 12 are: 1, 2, 3, 4, 6, and 12 of which only 2 and 3 are prime numbers. Prime numbers 2 and 3 are said to be prime factors of 12

Expressing a number as a product of its factors is called factorization. Say, if we write 12=3 x 4 then 12 has been factorized in terms 3 and 4.

Other factorizations of 12 are:

(i)12 = 6x2 =2x3x2

(ii)12 = 4x3 =2x2x3 

In all the above factorizations we always end up with one factorization i.e.

2x2x3 this is called prime factorization.

Conclusively, it can be  stated that, when a natural number is expressed as a product of its prime factors then the factorization of the number is called complete prime factorization.

Other examples of prime factorization:

(i) 24=2x2x2x3

(ii) 9=3x3

(iii) 14=2x7

(iv) 18= 2x3x3 and so on. 

Each of the above is a prime factorization for the given number.

NUMBERS AND DIVISIBILITY

DIVISIBILITY TEST OF NATURAL NUMBERS

DIVISIBILITY BY 2:

A number is divisible by two if its  last digit is divisible by 2 i.e. if its last digit is  even i.e. 0, 2, 4, 6 or 8

For example 100, 144, 186, 198 and 10004 are divisible by 2

DIVISIBILITY BY 3:

A number is divisible by 3 if the sum of its digits is divisible by three  

For example 300 is divisible by three because the sum of the digits  3 + 0 + 0 = 3 is divisible by 3

207 is divisible by three because  the sum of the digits 2 + 0 + 7 = 9   is divisible by 3.

DIVISIBILITY BY 4:

A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

For example  4548 is divisible by 4 because the last two digit, 4 and 8 sum up to  a multiple of 4 i.e. 4 + 8 = 12 which is a multiple of 4.

DIVISIBILITY BY 5:

A number is divisible by 5 if the last digit is 0 or 5.

For example 190, 35, 95, 1005 .etc…. are all divisible by 5 because ther last digit is either 0 or 5.

DIVISIBILITY BY 6:

A number is divisible by 6 if it is divisible b both 2 and 3.

For example 36 is divisible by both 2 and 3. So, it is divisible by 6.

DIVISIBILITY BY  8:

A number is divisible by 8 if the number formed by the last three digits is divisible by 8.

For example 1056 is divisible by 8 because  056 is divisible by 8.

 

DIVISIBILITY BY 9:

A number is divisible by 9 if the sum of its digits is divisible by 9.

For example 18 is divisible by 9 because 1 + 8 = 9 is divisible by 9.

261261 is divisible by 9 because 2 + 6 + 1 +2 + 6 + 1=18 is divisible by 9

DIVISIBILITY BY 10:

A number is divisible by 10 if its last digit is 0.

For example  290, 710, 3480, 200 are all divisible by 10.

COMMON FACTORS AND COMMON MULTIPLES

COMMON FACTORS AND COMMON MULTIPLES

(1)Given two numbers 12 and 18:

(i) F12 ={1, 2, 3, 4, 6 and 12}
(ii)F18 ={1, 2, 3, 6, 9 and 18}

note that the numbers 1, 2, 3 and 6 are factors common to both  F12
and F18. i.e. F12 n F18 ={1, 2, 3, 6}

refer to the figure below

 (2)Given two numbers 8 and 15:

(i)F8={1, 2, 4, 8}
(ii)F15={1, 3, 5, 15}

it can easily be seen that the common factor(s) of both 8 and 15
is 1 i.e F8 n F15={1}
refer to the figure below:

(3)Given two numbers 4 and 6:

The multiples of 4 are M4={ 4, 8, 16, 20, 24, 28, 32, 36,...., 4n}
The multiples of 6 are M6={6, 12, 18, 24, 30, 36, 42,...., 6n}
of the are above multiples 12, 24, 36, ...are the common multiples
of 4 and 6 i.e. M4 n M6 ={12, 24, 36}

FACTORS AND MULTIPLES

FACTORS, MULTIPLES AND DIVISORS

When two or more numbers are multiplied together the result is called the product that is to say given two natural  numbers  A and B  their product is  C or C=AXB. 

for example 

(i)The product of 4 and 3 is 12 i.e. 4x3=12.

(ii)The product of 2, 5 and 6 is  60 i.e. 2 x 5 x 6 = 60.
A and B are called factors of the product C, and C ,the product, is called the multiple of each of the factors A or B
So, for  4x3=12,  4 is a factor of 12 and 12 is a multiple of 4.Similarly 3 is a factor of 12 and 12 is a multiple of 3

Thus a natural number that divides another number exactly is called a factor (or a divisor in the case of division) of that number, and the given number is called a multiple of that number.
Hence, A is a factor of C because on dividing  C by A, the remainder is zero

for example since
(i)4x3=12 and 12x1=12, 1, 3, 4 and 12 are all factors (or divisors) of 12 and 12 is a multiple of each of the numbers 1, 3, 4, and 12.
(ii)2x4=8 2x2x4=8, and 1x8=8, 1, 2, 4, and 8 are all factors of 8 and 8 is a multiple of each of the number 1, 2, 4, and 8.

Similary
18 has factors f18={1, 2, 3, 6, 9, 18}.

42 has factors f42={1, 2, 3, 6, 7, 14, 21, 42}.

2 has multiples M2={2, 4, 6, 8, 10,...., 2n} where n is an integral number. 

6 has multiples M6={6, 12, 18, 24,...., 6n} where n is an integral number. 

from the above results we can conclude that :

(i)one is a factor of every number.

(ii)The only factor of one is one itself.

(iii)Every natural number is a factor of itself.

(iv)Every number has at least two factors, namely 1 and itself.

(v)Every factor of a number is less than or equal to that number.

(vi)Every number has a finite number of factors.

(vii)Every number is a multiple of itself.

(viii)Every multiple of a number is greater than or equal to the  and  number 
itself.

(ix)Every number has an infinite number of its multiples.

PRIME NUMBERS AND THE NUMBER LINE

Every composite number can be expressed as a product of primes, and this factorization is unique, regardless of the  order in which the prime factors occurs.A prime number is a natural number which is not divisible by any other number except 1 and itself..So, 2, 3, 5, 7, 11, 13, 17,1 9,23 etc... are prime numbers.

A composite number  is any natural number divisible by any other number besides 1 and itself. So, 4, 6, 8, 9, 10, 12, 15, 16, ....etc. are composite numbers.

The Primality test
Let n be any natural number. To establish establish whether n is a prime or not the following steps are taken.
 

Step 1 : Find approximate square root of the  given number.

Step 2 : Divide the given number by prime numbers less than approximate square root of number. If given number is not divisible by any of these prime numbers, the number is a prime else it’s a composite.

Ex. Is 37 it a prime number ?

Sol.

Let n=37

Step 1 :  √n=√37 = 6.08 ≈ 7

Step 2 : Prime numbers < 7 are 2,3, and 5  all do not evenly divide 37. so, 37 is a prime number.


NUMBER LINE

INTEGERS ON A NUMBER LINE
Integers can be represented by points on a straight line known as the number line. For this, a straight line that extends infinitely in both directions(left and right) is drawn. A point O called the origin is marked on to the line, and points 1, 2, 3, 4…  and points -1, -2, -3, -4,…. are marked on the right and left hand side respectively of point O as shown in the figure below.

In this way, each positive integer is a point on the right hand side of point O and each negative integer is a point on the left hand side of point O on the number line.



RATIONAL NUMBERS ON A NUMBER LINE

By the same token, rational numbers can be represente as shown below