Table of contents 
Prime Numbers 
Composite Numbers 
Divisibility Rules 
Factors and Multiples 
Highest Common Factor H.C.F 
Lowest Common Multiple L.C.M 
Exponential Notation or Index Notation 
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The numbers which cannot be broken into actors other than 1 and itself are known as prime numbers, e.g., 2, 3, 5, 7, 11, 13,....
It is important to note that
 2 is the onl even number which is prime.
Numbers which can be expressed as product of factors other than 1 and itself are known as composite numbers, e.g., 4, 6, 8, 9, 10, 12, 14, 15,....
Note that composite numbers are not only even. The collection of composite numbers is a mixture of even numbers and odd numbers both.
4, 6, 8, 10, ...... are even composite numbers
9, 15, 21, ...... are odd composite numbers.
Since we cannot express 1 as a product of two different factors other than 1 and itself, we say that 1 is the only exceptional number which is neither prime nor composite.
To determine whether a particular number is prime or composite, we don't have a set formula. For this, we have to perform actual division so many times in order to check whether we can divide the given number by another number or not. A reek mathematician Eratosthenes developed a method, in the 3rd century B.C., to find prime numbers. His method for finding prime numbers is known as Sieve method. For numbers from 1 to 100, this method is explained below.
Continue keeping the next prime numbers and crossing out their multiples, until we are left with prime numbers only.
Thus, prime numbers from 1 to 100 are:
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.
Two Prime numbers which differ by 2 are called twin primes, e.g., (3, 5) (5, 7) (11, 13 ) (17, 19)
Such numbers will form a pair of consecutive odd numbers.
A set of three prime numbers differing by 2 form a prime triplet, e.g., (3, 5, 7)
(3, 5, 7) is the only prime triplet.
 A pair of two numbers which have no common actor other than 1 are called coprimes, e.g., (2, 3) (3, 4) (4, 5) (3, 7) (4, 9)
 Coprime numbers need not be rime numbers themselves, e.g, (4, 9)
The only condition for two numbers to be coprimes is that there is not even a single number, other than 1, which can divide both of them.
Divisibility rules help us to find whether a number divides another number completely without performing actual division.
When a number divides another number exactly, then the divisor is called a factor of the dividend.
The dividend is called a multiple of the divisor.
For example, 42 is divisible by 7, so 7 is a factor of 42 and 42 is a multiple of 7.
Consider, 105 = 7 × 15.
Here also, 7 and 15 are factors of 105 and 105 is a multiple of both 7 and 15.
Observe the following:
(a) All the factors of 15 are 1, 3, 5 and 15.
All the factors of 20 are 1, 2, 4, 5, 10 and 20.
Thus, common factors of 15 and 20 are 1 and 5.
(b) All the factors of 32 are 1, 2, 4, 8, 16, 32.
All the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
Thus, common factors of 32 and 48 are 1, 2, 4, 8, 16.
Edurev Tips: does not have to factors So, it is not a prime number. Also, does not have more than to factors. So, it is not a composite number. Hence, is neither a prime nor a composite number
In the example a given above, common factors of 15 and 20 are 1 and 5. Out of these two common factors, 5 is the highest greatest common factor. Therefore, we call 5 as the highest common factor of 15 and 20.
Again in example by, common factors of 32 and 48 are 1, 2, 4, 8 and 16. Out of these, 16 is the highest greatest common factor. Therefore, 16 is the highest common factor of 32 and 48.
Hence, the H.C.F. highest common factor of two or more than two numbers is the highest greatest among all their common factors.
Highest common factor H.C.F. is also called Greatest common divisor (G.C.D.).
Example 1: Find all the common factors of:
(a) 24 and 40
(b) 20 and 64
(a) 24 = 1 × 24, 24 = 2 × 12,
24 = 3 × 8, 24 = 4 × 6
Therefore, all the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
Also, 40 = 1 × 40, 40 = 2 × 20, 40 = 4 × 10, 40 = 5 × 8.
Therefore, all the factors of 40 are 1, 2, 4, 5, 8, 10, 20 and 40.
Thus, common factors of 24 and 40 are 1, 2, 4 and 8.
(b) 20 = 1 × 20, 20 = 2 × 10, 20 = 4 × 5
Therefore, all the factors of 20 are 1, 2, 4, 5, 10 and 20.
Also, 64 = 1 × 64, 64 = 2 × 32, 64 = 4 × 16, 64 = 8 × 8.
Therefore, all the factors of 64 are 1, 2, 4, 8, 16, 32 and 64.
Thus, common factors of 20 and 64 are 1, 2 and 4.
Example 2: Find the H.C.F. of 18, 45 and 63.
18 = 1 × 18, 18 = 2 × 9, 18 = 3 × 6
All the factors of 18 are 1, 2, 3, 6, 9 and 18.
Also, 45 = 1 × 45, 45 = 3 × 15, 45 = 5 × 9.
All the factors of 45 are 1, 3, 5, 9, 15 and 45.
Also, 63 = 1 × 63, 63 = 3 × 21, 63 = 7 × 9.
All the factors of 63 are 1, 3, 7, 9, 21 and 63.
Common factors of 18, 45 and 63 are 1, 3 and 9.
Thus, H.C.F. of 18, 45 and 63 is 9.
Example 3: Find the H.C.F. of 15 and 28.
15 = 1 × 15, 15 = 3 × 5.
All the factors of 15 are 1, 3, 5 and 15.
Also, 28 = 28 × 1, 28 = 2 × 14, 28 = 4 × 7
All the factors of 28 are 1, 2, 4, 7, 14 and 28.
Common factor of 15 and 28 is 1.
Thus, H.C.F. of 15 and 28 is 1.
We call 15 and 28 are coprime.
Edurev Tips: Two numbers are said to be coprime, if their H.C.F. is 1
Properties of H.C.F
Let us try to find out the multiples of 8 and 12.
Multiples of 8 are 8, 16, 24, 32, 40, 48, .....................
Multiples of 12 are 12, 24, 36, 48, 60, 72, .....................
Thus, common multiples of 8 and 12 are 24, 48, .....................
Again, find the common multiples of 10 and 15.
Multiples of 10 are 10, 20, 30, 40, 50, 60, .....................
Multiples of 15 are 15, 30, 45, 60, 75, .....................
Thus, common multiples of 10 and 15 are 30, 60, .....................
In the examples given above, common multiples of 8 and 12 are 24, 48, .... and common multiples of 10 and 15 are 30, 60, ....
The lowest common multiple of 8 and 12 is 24.
And the lowest common multiple of 10 and 15 is 30.
Hence, the lowest common multiple L.C.M. of two or more numbers is the least smallest of their common multiples.
Example 4: Find the first two common multiples of:
(a) 4 and 10
(b) 5, 6 and 15.
(a) Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, .....
Multiples of 10 are 10, 20, 30, 40, .....
Thus, the first two common multiples of 4 and 10 are 20 and 40.
(b) Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 60, 65, .....
Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, .....
Multiples of 15 are 15, 30, 45, 60, 75, .....
Thus, the first two common multiples of 5, 6 and 15 are 30 and 60.
Example 5: Find the L.C.M. of 12 and 18.
Multiples of 12 are 12, 24, 36, 48, 60, 72, .....
Multiples of 18 are 18, 36, 54, 72, 90, .....
Thus, common multiples of 12 and 18 are 36, 72, .....
Therefore, L.C.M. of 12 and 18 is 36.
Properties of L.C.M
When a number is multiplied with itself several times, we express the product in the form given below; called the exponential or index notation.
In 3^{2}, 3 is called the base and 2 is called the exponent or index or power.
Similarly in 6^{4}, 6 is the base and 4 is the exponent index.
Observe the following examples:
2 × 2 × 2 × 3 × 3 = 23 × 32
5 × 5 × 5 × 5 × 8 × 8 × 8 = 5^{4} × 8^{3}
10 × 10 × 10 × 10 × 10 × 17 × 17 × 17 × 17 = 10^{5} × 17^{4}
Example 6: Write in the exponential form:
(a) 5 × 5 × 5 × 5 × 5 × 5
(b) 8 × 8 × 8 × 8 × 9 × 9
(a) 5 × 5 × 5 × 5 × 5 × 5 = 5^{6}
(b) 8 × 8 × 8 × 8 × 9 × 9 = 8^{4 }× 9^{2}.
Example 7: Write in the product form:
(a) 8^{7}
(b) 6^{3} × 11^{4}
(a) 8^{7 }= 8 × 8 × 8 × 8 × 8 × 8 × 8
(b) 6^{3 }× 11^{4} = 6 × 6 × 6 × 11 × 11 × 11 × 11.
Edurev Tips: In index notation, we cannot interchange base and exponent, i.e., 2^{3} is not equal to 3^{2}.
Observe the following:
54 = 1 × 54, 54 = 2 × 27, 54 = 3 × 18, 54 = 6 × 9, 54 = 2 × 3 × 9,
54 = 2 × 3 × 3 × 3
We see that in 54 = 2 × 3 × 3 × 3, all the factors are prime.
Thus, prime factorisation of 54 is 2 × 3 × 3 × 3.
Similarly , prime factorisation of 24 is 2 × 2 × 2 × 3.
Prime factorisation of 45 is 3 × 3 × 5.
When a number is expressed as a product of prime numbers, we call it prime factorisation
To get the prime factorisation of a number, we divide the given number b the prime numbers 2, 3, 5, 7, 11, etc., so long as the quotient is divisible b that number.
The prime factorisation of a number can be shown diagrammatically, known as factor tree.
Example 8: Express 56 as the product of prime factors.
We proceed as follows:
Hence, the prime factorisation of 56 is
= 2 × 2 × 2 × 7 = 2^{3} × 7.
Example 9: Draw a factor tree to find the prime factorisation of 96.
Thus, the prime factorisation of 96 is
= 2 × 2 × 2 × 2 × 2 × 3 = 2^{5} × 3.
To find the H.C.F. of two or more numbers, we express each one of them as the product of prime factors. Then, the product of terms containing least powers of common factors is their H.C.F.
Example 10: Find the H.C.F. of 28 and 72 by prime factorisation method.
Resolving each of the given numbers into prime factors, we get
28 = 2 × 2 × 7 = 2^{2} × 7
72 = 2 × 2 × 2 × 3 × 3 = 2^{3} × 3^{2}
H.C.F. of 28 and 72 = Product of common prime factors
= Product of terms with smallest powers of common factors
= 2^{2} = 4
Thus, H.C.F. of 28 and 72 = 4.
Example 11: Find the H.C.F. of 14, 84 and 105.
Resolving the given numbers into prime factors, we get.
14 = 2 × 7
84 = 2 × 2 × 3 × 7 = 22 × 3 × 7
105 = 3 × 5 × 7
H.C.F. of 14, 84 and 105 = Product of common prime factors
= Product of terms with smallest powers of common factors.
= 7^{1} = 7
Thus, the H.C.F. of 14, 84 and 105 = 7.
To find the HCF of two or three given numbers, divide the greater number by the smaller one. Then divide the divisor by the remainder. Go on repeating the process of dividing the preceding divisor by the remainder, till zero remainder is obtained. The last divisor is the HCF of the given numbers.
Example 12: Find the H.C.F. of 144, 180 and 324.
We first find the HCF of any two numbers (144 and 180).
Since, last divisor is 36, so 36 is the HCF of 144 and 180.
Now, we find the HCF of 36 and 324.
Hence, HCF of 144, 180 and 324 is 36.
To find the L.C.M. of two or more numbers, we express each one of them as the product of prime factors. Then the product of terms with highest powers of all the factors gives their L.C.M.
Example 13: Find the L.C.M. of 15 and 40, using prime factorisation method.
Resolving the given numbers into prime factors, we get
So, 15 = 3 × 5
Also, 40 = 2 × 2 × 2 × 5 = 23 × 5.
The different prime factors of 15 and 40 are 2, 3 and 5.
L.C.M. of 15 and 40 = Product of terms containing highest powers of 2, 3 and 5
= 2^{3} × 3 × 5 = 8 × 3 × 5 = 120.
Example 14: Find the L.C.M. of 16, 48 and 64.
Resolving each of the given numbers into prime factors, we get
16 = 2 × 2 × 2 × 2 48 = 2 × 2 × 2 × 2 × 3, 64 = 2 × 2 × 2 × 2 × 2 × 2
= 2^{4} = 2^{4} × 3 = 2^{6}
The different prime factors of 16, 48 and 64 are 2 and 3.
Thus, L.C.M. of 16, 48 and 64
= Product of terms containing highest powers of 2 and 3.
= 2^{6} × 3
= 64 × 3 = 192.
We proceed as below:
Example 15: Find the LCM of 18, 36 and 42 by long division method.
So, LCM of 18, 36 and 42
= 2 × 3 × 3 × 2 × 7 = 252
Hence, LCM of 18, 36 and 42 = 252.
For two given numbers, we have.
Example 16: The H.C.F. of two numbers is 52 and their L.C.M is 312. If one of the numbers is 104, find the other.
It is given here that:
H.C.F. = 52, L.C.M. = 312 and one number = 104
The other number
Hence, the other number is 156.
Example 17: The product of two numbers is 867 and their HCF is 17. Find their L.C.M.
We know that:
Hence, the LCM of given numbers is 51.
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