Table of contents  
Introduction  
Factors  
Multiples  
Prime Numbers  
Composite numbers  
Tests for divisibility of numbers  
Highest Common Factor  
Lowest Common Multiple (LCM) 
Suppose 4 more children visit his place at the same time. Can he distribute 24 muffins equally among all children?
Yes, he can!!! 24 ÷ 12 = 2 i.e .24 = 12 x 2
From this calculation we can see that 24 can be written as a product of two numbers in different ways as 24 = 6 × 4; 24 = 8 × 3; 24 = 12 × 2.
This means 2, 3, 4, 6, 8 and 12 are exact divisor of 24. They are known as factors of 24.
A factor of a number is defined as the number which is an exact divisor of that number.
When two (or more) numbers have the same factor, that factor is known as a common factor.
Let’s find put common factors of 6 and 18.
6 = 1 x 6, 6 = 2 x 3, therefore, factors of 6 = 1, 2, 3, 6 18 = 1 x 18,
18 = 2 x 9, 18 = 3 x 6, therefore, factors of 18 = 1, 2, 3, 6, 9, 18.
The numbers which appear in both the list are: 1, 2, 3, 6.
Hence, common factors of 6 and 18 are 1, 2, 3 and 6.
Example: Write all the factors of the following numbers:
(a) 23
(b) 36
Sol: (a) 23 = 1, 23
(b) 36 = 1 x 36,
36 = 2 x 18,
36 = 3 x 12,
36 = 4 x 9
36 = 6 x 6
Therefore, factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
A perfect number is defined as a number for which sum of all its factors is equal to twice the number.
For eg: (i) The factors of 6 are 1, 2, 3 and 6,
1 + 2 + 3 + 6
= 12
= 2 x 6.
Therefore, 6 is a perfect number.
(ii) The factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248
1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 + 496
= 992
= 2 x 496.
Therefore, 496 is a perfect number.
A multiple of a number is any whole number obtained by multiplying that number by another whole number.
For example, the multiples of 3 are 3, 6, 9, 12, 15, 18 and so on.
Every multiple of a number is greater than or equal to that number. The number of multiples of a given number is infinite. Every number is a multiple of itself.
(i) Every number is a multiple of 1.
For eg: 7 x 1 = 7,
9 x 1 = 9.
(ii) Every number is the multiple of itself.
For eg: 1 x 7 = 7
1 x 21 = 21
(iii) Zero (0) is a multiple of every number.
For eg: 0 x 9 = 0,
0 x 11 = 0.
(iv) Every multiple except zero is either equal to or greater than any of its factors.
For eg: multiple of 7 = 7, 14, 28, 35, 77, …………., etc.
(v) The product of two or more factors is the multiple of each factor.
For eg: 3 x 7 = 21. So, 21 is the multiple of both 3 and 7.
(vi) There is no end to multiples of a number.
5, 10, 15, 20, 25, …………….., 100, 105, 110, …………………., are the multiples of 5.
Example: List the first 5 multiples of 5, 8 and 9.
Sol: First 5 multiples of 5 are : 10, 15, 20, 25, 30
First 5 multiples of 8 are : 8, 16, 24, 32, 40
First 5 multiples of 9 are: 9, 18, 27, 36, 45
The Common multiples of two or more numbers are the multiples that are common to every given number.
Let’s find the common multiples of 4 and 6.
Example: Find all the multiples of 9 upto 100.
Sol: Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99.
The numbers other than 1 whose only factors are 1 and the number itself are called prime number.
For eg: 5 = 1 x 5, 23 = 1 x 23
Any set of numbers which do not have any other common factor other than 1 are called coprime or relatively prime numbers.
For example: Factors of 5 = 1, 5 Factors of 6 = 1, 2, 3, 6. This shows that 5 and 6 have no common factor other than 1. Therefore, they are coprime numbers.
Properties of coprime numbers
Twin primes are a pair of primes which differ by 2.
First few twin primes are E.g. (3, 5); (5, 7); (11, 13); (17, 19); (29, 31); (41, 43)
Example : Express 44 as the sum of two odd primes.
Sol: 44 = ____ + ____.
Here we have to find 2 numbers which are odd as well as prime numbers and whose sum is 44.
Odd prime numbers upto 44 are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43.
Now let’s find out a pair of numbers whose sum is 44.
Sum of 3 and 41 is 44. So, 44 = 3 + 41.
A composite number is defined as a number having more than two factors.
For eg: (i) Factors of 9 are 1, 3 and 9. Therefore it is a composite number.
(ii) Factors of 14 are 1, 2, 7 and 14. Therefore it is a composite number.
Note:
(i) 1 is neither prime nor composite number.
(ii) Any whole number greater than 1 is either a prime or composite number.
Example 1: Write seven consecutive composite numbers less than 100 so that there is no prime number between them.
Sol: 89 and 97 are the prime numbers. 90, 91, 92, 93, 94, 95, 96 lies in between 89 and 97. All these numbers have more than 2 factors. Therefore, all the numbers are composite numbers.
Example 2: Write down separately the prime and composite numbers less than 20
Sol: (i) Prime Numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, 19
(ii) Composite Numbers less than 20 are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18.
Example: A number is divisible by both 5 and 12. By which other number will that number be always divisible?
Sol: The number is divisible by 5 and 12.
Since 5 and 12 are coprime numbers so the number must be divisible by the product 5 × 12 = 60.
So, the given number will always be divisible by 60.
(HCF) of two or more given numbers is the highest (or greatest) of their common factors. It is also known as Greatest Common Divisor (GCD).
To find the HCF of two or more numbers, we can use any of the following method.
(i) Common factor method
(ii) Prime factorization method
Let’s find the HCF of 27 and 45.
Ques: Find the HCF of the following numbers. 18, 54, 81
Sol: Let’s find the HCF by prime factorization method. 18 = 2 x 3 x 3
54 = 2 x 3 x 3 x 3
81 = 3 x 3 x 3 x 3
Multiply all the factors which appear in both the list. i.e 3 x 3 = 9.
The HCF of 18, 54 and 81 is 9.
Example: LCM of 7 and 13.
7 = 7 x 1
13 = 1 x 13.
So, LCM is 7 x 13 = 91.
Let's find out lowest common multiple of the numbers 8, 12 and 18.
This method works only when there are very small numbers.
Let's find out the LCM of 90, 100 and 150.
A very convenient method to find the LCM is the common division method.
In this method of prime factorisation we proceed as follows:
Example 1 : Find the L.C.M. of the numbers 36, 48 and 72.
Sol: LCM = 2 x 2 x 2 x 2 x 3 x 3 = 144
Example 2: Find the least number which when divided by 18, 28, 32 and 42 leaves a remainder 5 in each case.
Sol: First we should find the LCM of 18, 28, 32 and 42.
LCM = 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7 = 2016
2016 is the least number which when divided by the given numbers will leave remainder 0 in each case. But we need the least number that leaves remainder 5 in each case.
Therefore, the required number is 5 more than 2016 i.e. 2016 + 5 = 2021.
9 videos114 docs49 tests

1. What are factors in mathematics? 
2. What are prime numbers? 
3. How can we find the Highest Common Factor (HCF) of two numbers? 
4. What is a composite number? 
5. How can we determine if a number is divisible by another number without actually dividing it? 

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