• Skip to main content
  • Skip to secondary menu
  • Skip to primary sidebar
  • Skip to footer
  • ICSE Solutions
    • ICSE Solutions for Class 10
    • ICSE Solutions for Class 9
    • ICSE Solutions for Class 8
    • ICSE Solutions for Class 7
    • ICSE Solutions for Class 6
  • Selina Solutions
  • ML Aggarwal Solutions
  • ISC & ICSE Papers
    • ICSE Previous Year Question Papers Class 10
    • ISC Previous Year Question Papers
    • ICSE Specimen Paper 2021-2022 Class 10 Solved
    • ICSE Specimen Papers 2020 for Class 9
    • ISC Specimen Papers 2020 for Class 12
    • ISC Specimen Papers 2020 for Class 11
    • ICSE Time Table 2020 Class 10
    • ISC Time Table 2020 Class 12
  • Maths
    • Merit Batch

A Plus Topper

Improve your Grades

  • CBSE Sample Papers
  • HSSLive
    • HSSLive Plus Two
    • HSSLive Plus One
    • Kerala SSLC
  • Exams
  • NCERT Solutions for Class 10 Maths
  • NIOS
  • Chemistry
  • Physics
  • ICSE Books

Finding Factors And Multiples

February 9, 2023 by Veerendra

FACTORS AND MULTIPLES

Finding Factors And Multiples 1
Mr Sharma wanted to withdraw Rs. 1000 from his bank account to purchase books for his children. The cashier gave him 10 hundred-rupee notes, i.e.,
Rs. 10 × 100 = Rs. 1000
Mr Sharma got the required amount. But the cashier could also give the same amount in the following ways:
Finding Factors And Multiples 2
Here, we observe that in each case Mr Sharma got the same amount of Rs. 1000. These numbers 1, 2, 5, 10, 20, 50, 100, 200, 500, and 1000 are factors of 1000. Hence, 1000 is a multiple of these numbers.
Here we will discuss only the natural numbers, that is positive integers.
If a = b × c, we say b and c are factors of a and a is a multiple of c and b.

Factors

Factor: A number which divides a given number exactly (without leaving any remainder) is called a factor of the given number.
Example: Factors of 12
12 = 1 × 12
12 = 2 × 6
12 = 3 × 4
Here, 1, 2, 3, 4, 6, and 12 are factors of 12.
finding-factors-multiples-2

Properties of Factors

  1. Every non-zero number is a factor of itself.
    Examples: 5 is a factor of 5. (5 ÷ 5 = 1)
    12 is a factor of 12. (12 ÷ 12 = 1)
  2. 1 is a factor of every number.
    Examples: 1 is a factor of 5. (5 ÷ 1 = 5)
    1 is a factor of 12. (12 ÷ 1 = 12)
  3. Every non-zero number is a factor of 0.
    Example: 5 and 12 are factors of 0 because
    0 ÷  5 = 0 and 0 ÷ 12 = 0
  4. The factors of a number are finite.

Multiples

Multiple: A multiple of any natural number is a number formed by multiplying it by another natural number.
Example: Multiples of 6 are 6 × 1 = 6; 6 × 2 = 12; 6 × 3 = 18; 6 × 4 = 24
Here, 6,12,18,24 are multiples of 6.
Example: Let us find the LCM and HCF of 24 and 36.
Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
Here, the highest common factor is 12.
∴ HCF = 12
Multiples of 24 = 24, 48, 72, 96,…
Multiples of 36 = 36, 72, 108,…
Here, the lowest common multiple is 72.
∴ LCM = 72
finding-factors-multiples-3

Properties of Multiples

  1. Every number is a multiple of itself.
    Examples
    (a) 3 × 1 = 3; 3 is the multiple of 3
    (b) 7 × 1 = 7; 7 is the multiple of 7
  2. Every number is the multiple of 1.
    Examples
    (a) 1 × 3 = 3; 3 is the multiple of 1
    (b) 1 × 7 = 7; 7 is the multiple of 1
  3. The multiples of a number are infinite (unlimited).

Even numbers: A number which is a multiple of 2 is called an even number.
Example: 2, 4, 6, 8, 10,…

Odd numbers: A number which is not a multiple of 2 is called an odd number.
Example: 1, 3, 5, 7, 9, 11,…

Prime numbers: A number which is greater than 1, and has exactly two factors (1 and the number itself) is called a prime number.
Example: Factors of 2 = 1, 2
Factors of 3 = 1, 3
Factors of 5 = 1, 5
Factors of 7 = 1, 7
Factors of 11 = 1, 11
Here, 2, 3, 5, 7, 11 etc. are all prime numbers.

Composite numbers: A number, which is greater than 1 and has more than two factors is called a composite number.
Examples: Here,
Factors of 4 = 1, 2, 4
Factors of 6 = 1, 2, 3, 6
Factors of 8 = 1, 2, 4, 8
Factors of 9 = 1, 3, 9
Factors of 10 = 1, 2, 5

FINDING PRIME NUMBERS FROM 1 TO 100
We can find the prime numbers from 1 to 100 by following these steps (given by the Greek mathematician Eratosthenes).
Step 1: Prepare a list of numbers from 1 to 100.
Step 2: As 1 is neither prime nor composite number, cross it out.
Step 3: Encircle ‘2’ as a prime number and cross out all its other multiples.
Step 4: Encircle ‘3’ as a prime number and cross out all its other multiples.
Step 5: Encircle ‘5’ as a prime number and cross out all its other multiples.
Step 6: Continue this process till all the numbers are either encircled or crossed out.
Finding Factors And Multiples 3
All the encircled numbers are prime numbers and the crossed out numbers (except 1) are composite numbers.
Numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 are the prime numbers between 1 and 100.
This is called the ‘Sieve of Eratosthenes’.

Twin primes: Two prime numbers having a difference of 2 are known as twin primes.
Example: ( 3, 5 ), ( 5,7 ), (11,13 ), (17,19 ), etc are twin primes.

Co-primes: Two numbers are said to be co-primes if they have no common factor other than 1. In other words, two natural numbers are co-primes if their HCF is 1.
Example: ( 2, 3 ), ( 3, 4 ), ( 5, 6 ), ( 7, 8 ), and so on.

Example 1: Is 16380 a multiple of 28?
Solution: To check whether 16380 is a multiple of 28 or not, we have to divide 16380 by 28. If the remainder becomes zero, then it is a multiple of the number.
So, 16380 = 28 × 585, hence 16380 is a multiple of 28.

Example 2: Express 29 as the sum of three odd prime numbers.
Solution: 29 = 19 + 7 + 3
All 19, 7, and 3 are odd prime numbers.

DIVISIBILITY TESTS FOR 2, 3, 4, 5, 6, 7, 8, 9, 10, AND 11

If we want to know that a number is divisible by another number, we generally perform the actual division and see whether the remainder is zero or not. This process is time-consuming for division of large numbers. Therefore, to cut short our efforts, some divisibility tests of different numbers are given below.

Test of Divisibility by

Condition

Example

2

A number is divisible by 2, if its ones digit is 0, 2, 4, 6 or 8.1372, 468, 500, 966 are divisible by 2, since their ones digit is 2, 8, 0 and 6 respectively.

3

A number is divisible by 3, if the sum of its digits is divisible by 3.In 1881, the sum of digits is 1 + 8 + 8 + 1 = 18 which is divisible by 3. So 1881 is divisible by 3.

4

A number is divisible by 4, if the number formed by the last two digits is divisible by 4.30776, 63784, 864 are all divisible by 4. Since last two digits of the numbers, i.e., 76, 84, and 64 are divisible by 4.

5

A number is divisible by 5, if its ones digit is either 5 or 0.675, 4320, 145 all are divisible by 5 because their ones digit is 5 or 0.

6

A number is divisible by 6, if the number is divisible by 2 and 3.In 5922, ones digit is 2, so it is divisible by 2. The sum of digits in 5922 is 5 + 9 + 2 + 2 = 18, which is divisible by 3. So, 5922 is divisible by 6.

7

A number is divisible by 7, if the difference between twice the last digit and the number formed by other digits is either 0 or a multiple of 7.In number 2975, it is observed that the last digit in 2975 is 5. So, 297 – (2 x 5) = 287, which is a multiple of 7. Hence, 2975 is divisible by 7.

 8

A number is divisible by 8, if the number formed by its last three digits is divisible by 8.In 213456, the last three digits are 456 which is divisible by 8. So, the number 213456 is divisible by 8.

9

A number is divisible by 9, if the sum of its digits is divisible by 9.In 538425, the sum of the digits are (5 + 3 + 8 + 4 + 2 + 5) = 27 which is divisible by 9. So, 538425 is divisible by 9.

10

A number is divisible by 10, if the digit at ones place of the number is 0.The numbers 980, 63990 are all divisible by 10 because their ones digit is 0.

11

 

A number is divisible by 11, if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or a multiple of 11.In number 27896, the sum of the digits at odd places are (2 + 8 + 6) = 16. The sum of the digits at even places are (7 + 9 ) = 16. Their difference is 16 – 16 = 0. So, the number 27896 is divisible by 11.

Example 3: Test whether 72148 is divisible by 8 or not?
Solution: Here, the number formed by the last three digits is 148, which is not divisible by 8.
So, 72148 is not divisible by 8.

Example 4: Test whether 8050314052 is divisible by 11 or not?
Solution: The sum of the digits at even places = 8 + 5 + 3 + 4 + 5 = 25
The sum of digits at the odd places = 0 + 0 + 1 + 0 + 2 = 3
Difference = 25 – 3 = 22 22 is divisible by 11.
So, the number 8050314052 is divisible by 11.

Maths

Filed Under: Mathematics Tagged With: Divisibility Test, Factors, Factors And Multiples, Finding Prime Numbers, Maths, Multiples, Pre Algebra, Prime numbers, Properties of Factors, Properties of Multiples

Primary Sidebar

  • MCQ Questions
  • RS Aggarwal Solutions
  • RS Aggarwal Solutions Class 10
  • RS Aggarwal Solutions Class 9
  • RS Aggarwal Solutions Class 8
  • RS Aggarwal Solutions Class 7
  • RS Aggarwal Solutions Class 6
  • ICSE Solutions
  • Selina ICSE Solutions
  • Concise Mathematics Class 10 ICSE Solutions
  • Concise Physics Class 10 ICSE Solutions
  • Concise Chemistry Class 10 ICSE Solutions
  • Concise Biology Class 10 ICSE Solutions
  • Concise Mathematics Class 9 ICSE Solutions
  • Concise Physics Class 9 ICSE Solutions
  • Concise Chemistry Class 9 ICSE Solutions
  • Concise Biology Class 9 ICSE Solutions
  • ML Aggarwal Solutions
  • ML Aggarwal Class 10 Solutions
  • ML Aggarwal Class 9 Solutions
  • ML Aggarwal Class 8 Solutions
  • ML Aggarwal Class 7 Solutions
  • ML Aggarwal Class 6 Solutions
  • HSSLive Plus One
  • HSSLive Plus Two
  • Kerala SSLC

Recent Posts

  • Notice Writing Class 10 ICSE Format, Examples, Topics, Exercises, Samples
  • Tum, Yushmad Ke Shabd Roop In Sanskrit – युष्मद् (तुम) शब्द के रूप – भेद, चिह्न उदाहरण (संस्कृत व्याकरण)
  • Advantages and Disadvantages of Media | List of Top 10 Media Advantages and Disadvantages
  • Provisional Certificate | Meaning, How Can We Get Provisional Certificate?
  • Happiness Essay | Essay on Happiness for Students and Children in English
  • Bahuvrihi Samas – बहुव्रीहि समास – परिभाषा, उदाहरण, भेद, सूत्र, अर्थ
  • Speech On Knowledge Is Power | Knowledge is Power Speech for Students and Children in English  
  • Who Inspires You Essay | My Biggest Inspiration Essay, Person Who Inspired Me Essay 
  • Dog Essay | Essay on Dog for Students and Children in English
  • Paragraph On Work Is Worship 100, 150, 200, 250 to 300 Words for Kids, Students And Children
  • 10 Lines on National Flag of India for Students and Children in English

Footer

  • RS Aggarwal Solutions
  • RS Aggarwal Solutions Class 10
  • RS Aggarwal Solutions Class 9
  • RS Aggarwal Solutions Class 8
  • RS Aggarwal Solutions Class 7
  • RS Aggarwal Solutions Class 6
  • Picture Dictionary
  • English Speech
  • ICSE Solutions
  • Selina ICSE Solutions
  • ML Aggarwal Solutions
  • HSSLive Plus One
  • HSSLive Plus Two
  • Kerala SSLC
  • Distance Education
DisclaimerPrivacy Policy
Area Volume Calculator