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TET Syllabus - Mathematics Study Materials in Number System (Part VII)

 

 Teachers Eligibility Test - Paper 2

 

Class – VI Mathematics

 

                                       NUMBER SYSTEM

   Prime Factorization:

                   Expressing a given number as a product of factors that are all prime numbers is called the prime factorization of a number.

For example, 36 can be written as product of factors as

                              36 =1 x 36;

                              36 = 2 x 18;

                              36 = 3 x 12;

                              36 = 4 x 9;

                              36 = 6 x 6

Here, the factors of 36 can be found easily as 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Note that not all the factors of 36 are prime numbers.

There are two prime factorization methods. They are

                           1. Division Method

                           2. Factor Tree Method

1. Division Method:

Find the prime factorization of 60.

                      

60 = 2x2x3x5

Hence the factors are 2x2x3x5.

 

2. Factor Tree Method:

Find the prime factorisation of 60.

                               

60 = 2 x 30 = 2 x 15 = 3 x 5

Hence the factors are 2x2x3x5.

Common Factors

                Consider the numbers 45 and 60. Use of divisibility tests will also help us to find the factors of 45 and 60.

 The factors of 45 are 1,3,5,9,15 and 45 and the factors of 60 are 1,2,3,4,5,6,10,12,15,20, 30 and 60.

Here, the common factors of 45 and 60 are 1, 3, 5 and 15.

Highest Common Factor (HCF)

              The Highest Common Factor HCF of two or more numbers is the highest number that divides the numbers exactly.

Example

Find the HCF of 63 and 42.

Solution:

              The prime factorisation of 63 = 3 x 3 x 7

               The prime factorisation of 42 = 2 x 3 x 7.

Then the common prime factors of 63 and 42 are 3 and 7

 And so the highest common factor is 3 x 7 =21.

    

·        The Highest Common Factor (HCF) is also called as the Greatest Common Divisor (GCD) or the Greatest Common Factor (GCF).

·        HCF (1, x) =1

·        HCF (x, y) = x, if y is a multiple of x.

For example, HCF (3, 6) = 3.

·        If the HCF of two numbers is 1, then the numbers are said to be co-primes or relatively prime.

Here, the two numbers can both be primes as (5, 7) or both can be composites as (14, 27) or one can be a prime and other a composite as (11, 12).

 

Example 3:

Find the HCF of the numbers 24 and 48 by division method.

Solution:

                  

The prime factorisation of 24 = 2x2x2x3

                                    

The prime factorisation of 48 = 2x2x2x2x3     

 

Here, the prime factorisation of 24 = 2x2x2x3

The prime factorisation of 48 = 2x2x2x2x3     

The product of common factors of 24 and 48 = 2 x 2 x 2 x 3 = 24

And so, HCF (24, 48) = 24.

Common Multiples:

Let us now write the multiples of 4 and 10.

Multiples of  are 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,…

Multiples of 10 are 10,20,30,40,50,60,70,80,90,100,…

Here, the common multiples of 4 and 10 are 40 and 60 and will go on without ending.

                     As multiples of a number are infinite, we can think of the Least Common Multiple of numbers, shortly denoted as LCM.

 

Least Common Multiple (LCM):

We can find the least common multiple of two or more numbers by the following methods.

               1. Division Method

               2. Prime Factorisation Method

Example:

Find the LCM of 144 and 198.

By Division Method

                       

LCM = product of all prime factors

           = 2 x 3 x 3 x 8 x 11

           = 1584

Thus, the LCM of 144 and 198 is 1584.

By Prime Factorisation Method

144 = 2 x 72

       = 2 x 3 x 24

       = 2 x 3 x 3 x 8

198 = 2 x 99

       = 2 x 3 x 33

       = 2 x 3 x 3 x 11

Now, LCM = (product of common factors) X (product of factors that are not                      common)

                   = (2 x 3 x 3) x (8 x 11)

                   = 18 x 88

                  = 1584

Thus, LCM of 144 and 198 is 1584.

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