## 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 12The factors of

**30**are: 1, 2, 3, 5, 6, 10, 15 and 30So 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 12The factors of

**30**are: 1, 2, 3, 5, 6, 10, 15 and 30So the common factors of 12 and 30 are:

*6 is the Highest number so it is HCF of 12 & 30.***1, 2, 3 and 6 ->**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