HCF and LCM

HCF(Highest Common Factor)

Common Factor: 

A number is said to be a factor of another numbers when it divides other numbers exactly.

Example:
a) 12 and 30
The factors of 12 are: 1, 2, 3, 4, 6 and 12
The factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30
So the common factors of 12 and 30 are: 1, 2, 3 and 6
 
b) 15, 30, 45
The Common Factors of 15, 30 & 45 are: 3, 5 & 15
 

HCF: 

HCF of two or more numbers is the greatest number that divides each of them exactly.  

H.C.F. is also called Greatest Common Divisor (G.C.D.)
Example: 

From the Above Common Factor Example we can find HCF by finding greatest element of the set of common factors

a) 12 and 30
The factors of 12 are: 1, 2, 3, 4, 6 and 12
The factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30
So the common factors of 12 and 30 are: 1, 2, 3 and 6 -> 6 is the Highest number so it is HCF of 12 & 30.
 
b) 15, 30, 45
The Common Factors of 15, 30 & 45 are: 3, 5 & 15 -> 15 is the Highest number so it is HCF of 15, 30 & 45.

Methods to Find HCF:

1. Factorization Method:

Example:  Find the HCF 24, 36 and 60 using Factorization Method.

2. Division Method:

Example: Find the HCF 24, 36 and 60 using Division Method.

HCF of Decimals:

Example: Find the HCF of 0.48, 0.72 and 0.108

 

HCF of Fractions: 

Example: Find the HCF of 2/3 , 3/5 , 4/7, 9/13

 

LCM(Least Common Factor)

Common Multilple: 

Common Multiple is number which is exactly divisible by each of the numbers under consideration.

Example:

Common Multiple of 3 & 5 are: 15, 30, 45, 60...

LCM:

LCM of two or more numbers is the least or smallest number which is exactly divisible by each of them.

Example:

a) 3 & 5

Common Multiple of 3 & 5 are: 15, 30, 45, 60... but smallest is 15 which is LCM of 3 & 5.

Methods to Find LCM:

1. Factorization Method:

Example: Find the LCM 24, 36 and 60 using Factorization Method.

2. Division Method:

Example: Find the LCM 24, 36 and 60 using Division Method.

LCM of Decimals:

Example: Find the LCM of 0.48, 0.72 and 0.108

 

LCM of Fractions:

Example: Find the LCM of 2/3 , 3/5 , 4/7, 9/13

Product of two numbers

First Number X Second Number = LCM X HCF

 

Co-primes

Two numbers are said to be co-primes if their HCF is 1.

Questions Asked from this Chapter:

 

Question Type 1: Based on ( LCM X HCF = First Number X Second Number )

 

The LCM of two numbers is 30 and their HCF is 5. One of the number is 10. The other is:
a) 20 
b) 25
c) 15
d) 5
 
The Product of two numbers is 216. If the HCF is 6, Then their LCM is:
a) 72
b) 60
c) 48
d) 36
 

Question Type 2: Based on Finding the LCM of the Numbers

 

Question Type 3: Based on Finding the HCF of the Numbers

 

Question Type 4: Based on the Ratio of the Numbers

 

The ratio of two numbers is 3:4 and their HCF is 4. Their LCM is:
a) 12
b) 16
c) 24
d) 48
 

Question Type 5: Based on the Applications of LCM and HCF

HCF and LCM of two numbers are 7 & 140 respectively. If the numbers are between 20 & 45, the sum of the numbers is:
a) 70
b) 77
c) 63
d) 56
 
The HCF of two numbers is 8. Which of the following can never be their LCM?
a) 24
b) 48
c) 56
d) 60

Question Type 6: Based on Addition, Difference, Multiplication and Division of the Numbers

 

The sum of two numbers is 45. Their difference is 1/9 of their sum. Their LCM is:
a) 200
b) 250
c) 100
d) 150
 
The product of two numbers is 4107. If the HCF of the numbers is 37, the greater number is:
a) 185
b) 111
c) 107
d) 101
 

Question Type 7: Based on the Pairs of the Numbers

The HCF and the product of two numbers are 15 and 6300 respectively. The number of possible pairs of numbers is:
a) 4
b) 3
c) 2
d) 1