7. L.C.M. and H.C.F.

 

HCF (OR GCD) AND LCM

Learn and Remember:

 1. HCF : The greatest among the common factors of the given numbers is called the HCF of those numbers.

e.g. factors of 8:1, 2, 4, 8; factors of 12 : 1, 2, 3, 4, 6, 12.

The common factors of 8 and 12 : 1, 2, 4.

The greatest among them : 4.

∴  the HCF of 8 and 12 is 4.

2. LCM : The lowest of the common multiples of the given numbers is called the LCM of those numbers.

e.g. multiples of 12 = 12, 24, 36, 48, 60, 72, ...

multiples of 18 = 18, 36, 54, 72, ..

The common multiples of 12 and 18 are 36, 72, ... .

The lowest of these common multiples is 36.

∴ the LCM of 12 and 18 is 36.

3. The HCF of two consecutive even numbers is 2.

e.g. the HCF of 16 and 18 is 2.

• The LCM of two consecutive even numbers is half their product.

e.g. the LCM of 16 and 18 = ½ x 16 x 18 = 144. (Check yourself)

4. For two consecutive odd numbers, the HCF is 1 and the LCM is their product.

e.g. the HCF of 11 and 13 is 1 and their LCM is 11 x 13= 143.

6. The HCF of two coprime numbers is 1 and their LCM is their product.

7. When one number is a factor of the other number, then the smaller number is the HCF and the greater number is the LCM of the two given numbers. e.g. for the numbers 12 and 24, 12 is the HCF and 24 is the LCM.

8. HCF = the product of common prime factors.

9. LCM = the product of common prime factors x the product of non-common prime factors.

∴  LCM = HCF x the product of non-common prime factors.

9. HCF x LCM = First number Second number.

10. The methods of finding HCF :

(1) Factorisation method

(2) Prime factorisation method

(3) Division method. Method

(1) is already discussed.

(2) Prime factorisation method :

Ex. Find the HCF of 12, 15 and 18.

12 = 2 x 2 x3

15 = 3 x 5

18 = 2 x3 x3

The common factor is 3.

HCF = 3.

(3) Division method :

Ex. Find the HCF of 12 and 18

 18÷ 12 =1    〈06               

 12÷6 = 2    〈0

      The last divisor is the HCF. Here, 6 is the HCF of 12 and 18.

11. The methods of finding LCM :

[A] Prime factorisation method (1) Horizontal method (2) Vertical method.

[B] Multiples method.

Vertical method : Find the LCM of 9, 12, 15.

3	9	12	15
3	3	4	5
2	1	4	5
2	1	2	5
5	1	1	5
	1	1	1

 LCM = 3 x 3 x 2 x 2 x 5 =9 x 4 x 5= 180.