Understanding factors and multiples is crucial in various mathematical concepts, such as prime factorization, least common multiple(LCM) and greatest common divisor (GCD). They are fundamental in solving problems involving fractions, simplification, and various algebraic manipulations.
Table of Contents
What are Factors ?
Factors are numbers that divide another number without leaving a remainder . For example , the factors of 12 are 1, 2, 3, 4, 6, 12.
or
Factors are the numbers that we multiply in order to get a product . 2 divides 6 exactly with remainder 0 . In other words , we say it as 2 is a factor of 6 .
What are Multiples?
Multiples are numbers that can be evenly divided by another number . For example multiples of 3 include 3, 6, 9, and so on , since they can be divided by 3 without leaving a remainder.
or
When a number divides another number exactly , then the dividend is called a multiple of the divisor.
Real Life Example :
Let’s take the number 15:
*Factors of 15 would be 1, 3, 5, and 15.
*Multiples of 15 would include 15, 30, 45, and so on.
In a practical situation , if you have a garden with rows of plants, you might choose to plant them in a group . The number of plant in each group could represent the multiples, while factors would indicate the ways you could count or arrange the plant in each row.
Properties of factors :
Property 1: 1 is a factor of every number as 1 divides every number exactly. Thus , it is a factor of every number. It is the smallest factor.
Property 2 : Every non-zero number is the largest factor of itself. 16 is the largest factor of 16. 34 is the largest factor of 34.
Property 3 : Every non-zero number is a factor of 0. 2 is a factor of 0 ,19 is a factor of 0.
Property 4 : Dividing by 0 is not defined . Therefore o is not a factor of any number.
Property 5 : Every factor of a non-zero number is either less than or equals to the given number . All the factor of 10 are 1, 2, 5, and 10 . Clearly, each one of them is less than or equals to 10.
Properties of multiples :
Property 1 : Every number is a multiple of 1 . Thus , multiples of 1 are ; 1, 2, 3, 4, …… 100, 101, ……. 10000 …….
Property 2 : Every non-zero number has unlimited number of multiples . Multiples os 5 are 5, 10, 15, ……. 100, …… 1000 …….. Clearly, 5 has unlimited number of multiples.
Property 3 : Every non-zero number is a multiples of itself . 5×1⇒5 Here 6 is a multiple of 6 . 23×1⇒23 Here 23 is a multiple of 23 etc.
Property 4 : 0 is a multiple of every non-zero number . We know that every non-zero number is a factor of 0.Therfore 0 is a multiple of every non-zero number.
Property 5: Every multiple of a non-zero number is either greater than or equal to the given number. Multiples of 6 are 6, 12, 18, 24, ……..Clearly, each one of them is greater than or equal to 6.
Property 6 : The first multiple of a number is the number itself. First multiple of 56 is 56, first multiple of 9 is 9 and so on .
Highest Common Factor (HCF) :
The HCF of some given numbers is the largest number that divides each one of them exactly. Let us see some examples.
Example : Find the HCF of 12 and 15 .
Factors of 12 are 1, 2, 3, 4, 6, and 12.
Factors of 15 are 1, 3, 5, and 15 .
Common factors of 12 and 15 are 1 and 3 ; Hence the HCF of 12 and 15 is 3 .
Lowest Common Multiples (LCM) :
LCM of some given numbers the smallest common multiple of those number.
Example : Find the LCM of 6 and 8 .
Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48 ……. Multiples of 8 are 8, 16, 24, 32, 40,48 ……
Common Multiples of 6 and 8 are 24 and 48. ; Hence the LCM of 6 and 8 is 24.
Differences between Factors and Multiples :
Factors : 1)Factors are defined as the exact divisors of the given number, 2) The number of factors is finite , 3) The operation used to find the factors is a division, 4) The outcome of the factors should be less than or equal to the giver number.
Multiples : 1)The multiples are defined as the numbers obtained when multiplied by other numbers , 2) The number of multiples are infinite, 3) The operation used to find the multiples is a multiplication, 4) The outcome of the multiples should be greater than or equal to the given number.
Practice Questions
- What is factor?
- What is multiples?
- What are the difference between factor and multiples?
Recommended Reading
What is Irrational Number Class 9 (With Examples.)
Difference Between Place Value and Face Value- (With 10 Solved Example).
FAQs :
Q.1 Find the factor of 15?
Answer – Factors of 15 are 1, 3, 5 and 15.
Q.2 Write the first 4 multiples of 10 ?
Answer – First 4 Multiples of 10 are 10, 20, 30, and 40.
Q.3 Write down the common multiples of 3 and 5 ?
Answer – The common multiples of 3 and 5 are 15, 30, 45 and so on.
