CBSE NCERT SOLUTIONS

Factors and Multiples

Case 1:Factors are the numbers that are multiplied to give  a product.

Example-5 x 4 = 20

5 and 4 are the factors of 20.

Pictorial representation:

Case 2: Factors of a number also divide that number without leaving any remainder.

Example:24 ÷ 3=8 or 24 ÷ 8= 3 , and here 8 and 3 are both the factor of 24.

II)When you multiple a number by 1,2,3,4,…, then the products are all multiples of the numbers.

Example: 3,6,9,12,… are all multiples of 3 as (3 × 1), (3 × 2), (3 × 3), (3 × 4),…

III)Rules of Divisibility:

The rules of divisibility will help you find which numbers devide others without leaving any reaminder.

Rule 1:

Rule 2:

Rule 3:

III)Prime Numbers and Composite Numbers:

What is a Prime Number?

The number which is having only two factors such as :

i)the number itself and

ii)the number 1

such number is known to be or called as prime number.

Examples:2,3,5,7

Defn: A prime number is any number greater than 1 that has only two factors –the number itself and the number 1

What is a Composite Number?

The number which is having more than two different factors is known to be or called as Composite number.

Examples:6

6 has 4 factors .6=3×2×1

But here the factors of 6 are 1,2,3,and 6.

Here 6 is the composite factors.

Defⁿ: A Composite number is any number greater than 1 that has only two factors –the number itself and the number 1.

What is a special number?

Then the next question is what about the number 1.

But as the rule of prime number It has two factors as the rule the number itself and the number 1.  And but here the number is same so 1 has only 1 factor.So it is not a prime number.

By the rule of composite number 1 has not more than one factors .so 1 is not a composite factor.

So 1 is a special number .It is neither prime nor composite.

What is twin primes?

When there are two prime numbers with difference of 2,then thy are called twin prime.

Examples: 3 and 5 are twin primes ,5 and 7 are twin primes.

Question:

Find all the twin primes from 1 to 100?

Prime factorization of a number:

When the factors of a number are all prime numbers ,it is called the prime factorisation of the number.

There are mainly 2 ways to find the prime factors of a given number:

a)Prime Factorisation Method

b)Division Method

a)Prime Factorisation Method:

Example:

56=2 × 28

     =2× 2× 14

    =2×2×2×7

So the prime factors of 56=2×2×2×7

b)Division Method:

So the prime factors of 56=2×2×3×13

What is Even number?

A number which is a multiple of 2 is called an even number.

All even numbers are end with the digits 0,2,4,6 or 8.

Example: 2,4 and so on

What is Odd number?

A number which is not a multiple of 2 is called an odd number.

All Odd numbers are end with the digits 1,3,5,7 or 9.

Example: 1,3 and so on.

HIGHEST COMMON FACTOR(HCF)

Defⁿ: The HCF of two numbers is the greatest number that can divide both numbers without leaving any remainders.

The HCF of two or more numbers is always smaller than or equal to the numbers.

Example:What is the HCF of numbers 12 and 18.

Process:

To Calculate HCF we have to calculate the common factors of the given numbers first.And to calculate the common factors of numbers we have to calculate the factors of the given numbers individually.

So ….these are the steps to calculate HCF :

Step 1: Calculate the Factors of the given numbers

Here ,the two given numbers are in this example are 12 and 18.So we have to calculate the factors of these 2 numbers first.

Factors of 12 : 1,2,3,4,6,12

Factors of 18 : 1,2,3,6,9,18

Step 2 : Calculate the common factors of the given numbers

Here ,the two given numbers are in this example are 12 and 18.

So the common factors are 1,2,3 and 6

Step 3 : Calculate the highest common factors from the common factors.

And in this example the highest common factor is 6.and here 6 is the greatest number that can divide both 12 and 18 without leaving any remainder.

Methods to find HCF of numbers:

There are three methods used to find the HCF of numbers:

1.Prime Factorisation Method

2.Common Division Method

3.Long Division Method

1.Prime Factorisation Method:

Example: Find the HCF of 24 and 32 using Prime Factorisation Method.

Step 1:Calculate first the prime factors of given numbers using any of the methods from Prime Factorisation Method and Division Method to find prime factors.

By Using Prime Factorisation Method:

24=2×12=2×2×6=2×2×2×3 

32=2×16=2×2×8=2×2×2×4=2×2×2×2×2 

So the prime factors of 24=2×2×2×3

So the prime factors of 32=2×2×2×2×2

By Using Division Method:

So the prime factors of 24=2×2×2×3

So the prime factors of 32=2×2×2×2×2

Step 2: Find the common prime factors from all the prime factors of numbers

Then the common prime factors of 24 and 32 are 2,2 and 2

Step 3: Then the HCF Of numbers is the multiples of common prime factors.

Then the HCF of 24 and 32 =2×2×2=8

2.Common Division Method:

Example: Find the HCF of 270 and 450 using Common Division Method.

Step 1:Divide the numbers by the lowest common prime number.Write the quotients below the respective numbers.

Step 2.Again now the quotient become the divident and now divide the divident by the lowest common prime number and write the new found quotients below the dividend.

Step 3.Stop this loop when there is no common prime number to devide by the newly formed dividend.

Pictorial representation :

So, here the prime factors of numbers are 2,3,3 ,5,3,5

So, here the common factors are 2,3,3 and 5

So,the HCF =product of all common factors=2×3×3×5=90

3.Long Division Method:

Find the HCF of 24 and 30 using the Long Division Method or continued division Method.

Step 1:Put the smaller number as the divisor and greater number as the dividend.

So here in this example we take 24 as divisor and 30 as the dividend.

Step 2: After dividing ,the remainder becomes the divisor and the previous case divisor becomes the dividend now.

Now 6 becomes the divisor and 24 becomes the dividend.

Step 3:Continue this process until you get the zero as the remainder.

Now the remainder is 0.So the process is stopped.

Step 4:The last divisor is the HCF .

Then the last divisor is 6.And then the HCF is 6.

Points to remember:

1)Products of an even number and any other number is always even.

2)Products of two odd numbers is always odd.

3)Numbers that have 1 as their HCF are called co-primes.

4)If a number is a factor of another number then their HCF is the smaller number .Example: HCF of 3 and 6 is 3.

Using HCF:

Example: Find the HCF of the numbers 63 and 48 which leave a remainder 3 in both numbers.

Here both the numbers leave 3 as the remainder .so the new numbers are the number substracted by 3 from both the numbers.

So the new numbers are 63-3=60 and 48-3=45.

So we have to find the HCF of these new numbers to get our answer.

HCF of 60 and 45 is 15.

So the HCF of the numbers 63 and 48 which leave a remainder 3 in both numbers is 15.

Points to remember:

1)Products of an even number and any other number is always even.

2)Products of two odd numbers is always odd.

3)Numbers that have 1 as their HCF are called co-primes.

4)If a number is a factor of another number then their HCF is the smaller number .Example: HCF of 3 and 6 is 3.