H.C.F of Two Numbers by Long Division Method.

HIGHEST COMMON FACTOR BY LONG DIVISION:

1) Long Division for two numbers:

Step-1: In the given two numbers, we need to divide the largest number (Dividend) with smallest number(Divisor) at once. The remainder (R₁) obtained in this step is the divisor for the next step.

Step-2: In this step, the Divisor in the step-1 is the dividend in the step-2 and R₁ is the Divisor. The remainder (R₂) obtained in this step is the Divisor for the Step-3.

Step-3:  The Divisor in the step-2, becomes the Dividend in this step. This Process of Division is continued till we get the Remainder ‘0’.

Step-4: This is the last step, in which we get the Remainder as ‘0’.The divisor in the last step is the Highest Common Factor.

For Bigger numbers we find H.C.F by using Long Division method.

Find H.C.F of two numbers by Long Division method:

1) 248 and 492.

Answer:

(i) We need to Divide largest number,Dividend= 492 with smallest number, Divisor= 248  at once.The Remainder in this step is  244,  which is the new Divisor for the second step.

(ii) Now, the Dividend is 248  and the Divisor is  244. In this step, after dividing the numbers at once we get the remainder as  4,  which is a new Divisorfor the next step.

(iii) Here, the Divisor is  4 and Dividend is 244. When we divide 244 with 4 at once we get the Remainder as ‘0’. By this we stop the division further. The Divisor: 4, in this step is the Highest Common Factor.(H.C.F).

   

Therefore,H.C.F of 248 and 492 = 2.

 2) 1584 and 1683.

Answer:

i) Dividend = 1683; Divisor = 1584 ; Remainder = 99.

ii) Divisor = 99 ; Dividend = 1584 ; Remainder = 0.

Here, we stop the division and the H.C.F = 99.

 

Therefore, H.C.F of 1584 and 1683 = 99.

 3) 1044 and 1512.

Answer:

(i) Dividend = 1512 ; Divisor = 1044 ; Remainder = 468.

(ii) Dividend = 1044; Divisor = 468 ; Remainder = 108.

(iii) Dividend = 468; Divisor = 108 ; Remainder = 36.

(iv) Dividend = 108 ; Divisor = 36 ; Remainder = 0. This is the last step.

Therefore, Highest Common Factor of 1044 and 1512 = 36.  

 

4) 396 and 420.

Answer:

(i) Dividend = 420; Divisor = 396; Remainder = 34.

(ii) Dividend = 396; Divisor = 34; Remainder = 22.

(iii) Dividend = 34; Divisor = 22; Remainder = 12.

(iv) Dividend = 22; Divisor = 12; Remainder = 10.

(v) Dividend = 12; Divisor = 10;  Remainder = 2.

(vi) Dividend = 10; Divisor = 2;  Remainder = 0.This is the end of the division.

Therefore, H.C.F of 396  and  420 = 2.

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.