Greatest Common Factors

GREATEST COMMON FACTOR ( G.C.F):

It is also called as Highest Common Factor( H.C.F) or Greatest Common Divisor( G. C.D).

Greatest Common Factor is the greatest factor of the numbers that divides the given numbers exactly.

If any common factor is not obtained for the numbers, then the G.C.F of the numbers is 1.

The G.C.F of numbers can be find out by any one of the methods. They are:

1) G.C.F by Listing out the Factors.

2)  G.C.F by using Factor-tree method.

3) G.C.F by using Short Division method.

The G.C.F obtained by any one of the above methods is Same.

 

1) G.C.F by Listing Out the Factors:

(i) We need to find the factors of the given numbers.

(ii)  The Common factors of the numbers are taken out.

(iii)  The greatest factor in those Common factors is the G.C.F.

 

 

Let the numbers be: 18 and 24.

Answer:

i) Factors of 18: 1, 2, 3, 6, 9, 18.

ii) Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.

Therefore, Common Factors of 18 and 24 : 1, 2, 3, 6.

Thus, the Greatest Common Factor(G.C.F) or H.C.F or G.C.D = 6.

 

2) G.C.F by Prime Factorization:

i) In this method, we need to find out the Prime factors of the given numbers either by Factor-tree method or Short division method.

ii) The common factors of the numbers are taken out.

iii) Those common factors obtained are multiplied, which gives the Greatest Common Factor.   

Let the numbers be : 18 and 24.

Answer:

Factors of 18:   Prime Factors of 18      = 2 * 3* 3	Factors of 60: Prime Factors of 60        = 2*2*3*5
Common factors of 18 and 60 = 2, 3. Greatest Common factor = 2*3 = 6.

                                                     

 

3) G.C.F by Short Division method:

i) The Prime factors of the numbers are found out by using Short division method.

ii) The common factors of the numbers are taken out.

iii) We need to multiply these common factors of the numbers, which gives Greatest Common Factors (G.C.F) of the given numbers.

 

Let the numbers be : 18 and 24.

Answer:

Factor of 18                                           Prime Factor of 18           =  2*3*3	Factor of 24                                                                                         Prime Factor of 24            = 2*2*2*3
Common Factors of 18 and 24 = 2, 3. Greatest Common Factor (G.C.F) of 18 and 24   = 2*3 = 6.

 

Find the Highest Common Factor:

1) 24 , 36 and 60.

Answer:

Finding H.C.F  by using  Factor-tree method:

 

 

Factors of 24 Factors of 24  = 2*2*2*3	Factors of 36 Factors of 36 = 2*2*3*3	Factors of 60 Factors of 60 = 2*2*3*5
Common Factors of 24, 36, and 60 = 2 , 2, 3. Highest Common Factor = 2 * 2* 3 = 12.

 

2) 18 and 25.

Answer:

Finding G.C.F by Listing out Factors method.

i) Factors of 18 =  1 , 2, 3, 6, 9, 18.

ii) Factors of 25 =  1, 5 ,25.

Therefore, Common factors of 18 and 25 = 1.

Here, Common factors are not their except factor 1.

Thus, the G.C.F of 18 and 25 = 1.

 

3) 16 and 30

Answer:

i) Factors of 16 = 1, 2, 4 , 8 ,16.

ii) Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30.

Therefore, Common factors of 16 and 30 =  1, 2.

Thus,Greatest Common Divisor ( G.C.D) =  2.

 

4) 36 , 54 and 63.

Answer:

i) Factors of 36 =  1, 2, 3, 4, 6, 9, 12, 18, 36.

ii) Factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54.

iii) Factors of 63 = 1, 3, 7, 9, 21 ,63.

Therefore, Common Factors of 36, 54 and 63 =  1, 3, 9.

Thus, Greatest Common Factor (G.C.F) = 9.

 

Common Factors

COMMON FACTORS:

Factors that, two or more numbers have in common are called common factors.

Find the Common factors of the numbers:

1) 12 and 36.

Answer:

We need to find the factors of the two given numbers separately and have to find the factors that are in common in both the numbers.

12 = 1*12 =  2*6 = 3*4.

36 = 1*36 = 2*18 = 3*12 = 4*9 = 6*6.

(i) Factors of number 12:  1, 2, 3, 4, 6, and 12.

(ii) Factors of number 36: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Here, the factors that are common in both the given numbers are:

Common factors: 1, 2, 3, 4, 6, and 12.

 

2) 18 and 57.

Answer:

18 = 1*18 = 2*9 = 3*6.

57 = 1*57 = 3*19.

(i) Factors of number 18:  1, 2, 3, 6, 9, and 18.

(ii) Factors of number 57: 1, 3, 19, and 57.

Therefore, Common factors: 1, 3.

 

3) 10, 25 and 60.

Answer:

10 = 1*10 = 2*5.

25 = 1*25 = 5*5.

60 = 1*60 = 2*30 = 3*20 = 4*15 = 5*12 = 6*10.

(i) Factors of number 10: 1, 2, 5, and 10.

(ii) Factors of number 25:  1, 5, and 25.

(iii) Factors of number 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

Therefore, Common factors : 1,  5.

4) 12, 18 and 36.

Answer:

12 = 1*12 = 2*6 = 3*4.

18 = 1*18 = 2*9 = 3*6.

36 = 1*36 = 2*18 = 3*12 = 4*9 = 6*6.

(i) Factors of number 12:  1, 2, 3, 4, 6, and 12.

(ii) Factors of number 18: 1, 2, 3, 6, 9, and 18.

(iii) Factors of number 36: 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Therefore, Common factors: 1, 2, 3, 6.

Prime Factorization

PRIME FACTORIZATION:

Writing a number as a product of its factors is called Factorization.

If we write a number as a product of its Prime factors, it is called Prime Factorization.

There are two methods to find the Prime factors of a number. They are:

(i) Factor tree method

(ii) Short division method.

(i) FACTOR TREE METHOD:

A factor tree shows the prime factors of a composite number in a “tree-like” form.

·        In this method, we factorize the numbers in such a way that at least one of the factors is prime factor.

·        We then factorize the composite factor further to get at least one prime factor.

·        We continue this way until all the factors we get are prime factors.

·        We can make different factor trees to find the same prime factorization.

1) 100.

Answer:

The two different factor trees for number 100:

 

Here, the factors trees gives same Prime Factorization.

Prime factors of number 100 : 2 * 2 * 5 * 5.

 

2)  72.

Answer:

The different factor trees are:

 

The different factor trees gives same Prime Factorization.

The prime factors of number 72  are : 2* 2* 2 * 3* 3.

3) 60

Answer:

The different factor trees are:

The different factor trees gives same Prime Factorization.

The Prime factors of number 60  are :  2* 2 * 3 * 5.

(ii) SHORT DIVISION METHOD:

We follow the below steps to find factors in this method. We have to use only Prime numbers to divide the number.

Step-1: First we divide the number by the smallest prime number which divides the number exactly.

Step-2: We divide the obtained quotient in step-1 again by the smallest or the next smallest prime number if it is not exactly divisible by the smallest prime number.

We repeat the process again and again till the quotient becomes 1.

Step-3: We multiply all the prime factors obtained, which gives the Number itself.

 

1) 32

Answer:

Step-1 : The number 32 is dividend. Dividing 32 with the smallest Prime number : 2, which divides exactly and leaves quotient as 16.

Step-2 : Now the new dividend is 16, which is divided exactly by the smallest prime number 2  again. This division process is continued till we get the quotient as 1.

Step-3: We multiply all the prime factors obtained, which gives the Number itself.

The factors of  32 are: 2 * 2 * 2* 2* 2.

 

2) 45.

Answer:

Step-1: The smallest prime number is 2. But it cannot divide 45 exactly. So, we go for the next smallest prime number, 3 which divides 45 exactly.

Step-2: The new dividend is 15, which again divided by the smallest prime number. This process is to be continued till quotient 1.

Step-3 : We multiply all the prime factors obtained, which gives the Number itself.

 

The factors of  45  are:  3 * 3 * 5.  

Prime and Composite Numbers.

PRIME AND COMPOSITE NUMBERS:   

Factors are either composite numbers or prime numbers.

PRIME NUMBER:

A prime number has only two factors, one:’1’ and number itself.

It cannot be divided evenly by any other numbers. Here's a list of prime numbers up to 100. These Prime numbers cannot be factored any further.

PRIME NUMBERS up to 100

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

COMPOSITE NUMBER:

A composite number is any number that has more than two factors.

Here is a list of composite numbers up to 20.The composite numbers can be factored further.

For example: 4 = 2* 2 ; 6= 3 * 2 ; 8 = 4 * 2; and so forth.

COMPOSITE NUMBERS up to 20

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20.

UNIQUE NUMBER:

The number ‘1’, is neither prime nor composite number. It is called a Unique number. Unique number has only one factor.

CONSECUTIVE PRIME NUMBERS:

The prime numbers which come one after the other are called Consecutive Prime Numbers.

PRIME NUMBERS up to 100 :  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

From the above list of Prime numbers up to 100: Only 2 and 3 comes one after other. So, they are Consecutive Prime Numbers.  

TWIN PRIMES:

Two Prime numbers which have a composite number in between are called Twin Primes.

The two Prime numbers 3 and 5, which have composite number,4  in between. Thus Prime numbers 3 and 5 are called Twin Primes.

The pair of Twin primes are: 3 and 5 ; 5 and 7; 11 and 13; 17 and 19;29 and 30; 41 and 43; 59 and 60;  71 and 73 .