## Factor and Multiple

If a divides b, then a is known as factor of b and b is known as multiple of a.

e.g if 12 divides 48, then 12 is a factor of 48 and 48 is a multiple of 12.

**Highest Common Factor (HCF)**

The greatest number which divides all given two or more numbers, is called HCF of given numbers.

e.g HCF of two numbers 28 and 42 iş 14 because 14 is greatest number that

divides 28 and 42 exactly.

**Methods to find HCF of given numbers**

There are two methods to find HCF of given two or more numbers.

**(1) Prime Factorisation Method**

In this method, find all prime factors of each given number. Find the product of all common prime factors. This product is required HCF of given numbers.

**Ex. Find the HCF of 36 and 84.**

A. 12

B. 4

C. 6

D. 18

**sol:-**

36=2 x 2 x 3 x 3

& 84=2 x 2 x 3 x 7

HCF of 36 and 84 = 2 x 2 x 3 = 12

**(2) Division Method**

In this method, first find the HCF of first two numbers by dividing larger number by smaller number.

Now find the the remainder. Divide the smaller number (divisor) by remainder and proceed with this step till we get the remainder zero.

The last divisor is required HCF of two numbers. In case of more than two numbers, find the HCF of third number with HCF of first two numbers.

**Ex- What is the square root of HCF of 36,40,48?**

A. 16

B.25

C.49

D.36

** Sol-** 36 )40( 1

4 )36( 9

36

———————————-

x

———————————-

HCF of 36, and 40 = 4

Now, HCF of 4 and 48

4 )48( 12

4 )4 (

—————————

8

8

———————————-

x

———————————-

HCF of 36,40 and 48 =4

Square of HCF =4×4= 16

**Least Common Multiple (LCM)**

The least number which is divisible by two or more given numbers, is called LCM of given numbers.

e.g. LCM of 28 and 42 is 84 because 84 is the least number which is divisible by both 28 and 48.**Methods to find LCM of given numbers**

There are two methods to find LCM of given numbers.

**(i) Prime Factorisation Method**

In this method, find the prime factors of all given numbers.

Find the product of the highest powers of all the factors

that occur in the given numbers. This product is required

LCM of given numbers.

**Ex. Find the LCM of 48,72 and 108.**

(a) 432

(b) 216

(c) 320

(d) 540

**Sol:**

**(ii)Division Method**

In this method, we put all numbers in a line separating them by commas. Divide these numbers by possible prime factors which exactly divides two or more numbers at same time and write quotient in next line and divide these quotients by prime numbers and follow this process unti we get quotient ‘1’. Now find the product of all prime factors. This product is required LCM of given numbers.

**Ex. Find the LCM of 24,30 and 42 .**

(a) 480

(b) 840

(c) 320

(d) None of these

**Require L.C.M of 24, 30 and 42**

**2x2x2x3x5x7=840**

**Note÷ LCM is divisible by HCF.**