A number is defined as an arithmetical value, expressed by a word, symbol, and figures.
These numbers can be written in one digit, two digits, threedigits in the generalized form.
Factors
A
factor of a number is an exact divisor of that number.
Eg:
Factor of 6 = 1,2,3,6
What
are the properties of factors?
 1 is a factor of every number.
 Every number is a factor of itself.
 Every number is a multiple of itself.
 Every factor of a number is an exact divisor of that number.
 Every factor of a number is less than or equal to that number.
 The factors of a given number are finite in number.
 The multiples of a given number are infinite in number.
 Every multiple of a number is greater than or equal to that number.
Perfect number
If the sum of all the factors of a number is equal to twice the number, then such a number is called a perfect number.
Eg: 28
Factors of 28 = 1,2,4,7,14 and 28
Sum = 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × 2
Multiple
Multiple of a number is the numbers obtained by multiplying that numbers with various Natural numbers.
Ex: 7
7 x 1 = 7
7 x 2 = 14
7 x 3 =
21
What are the Properties of Multiple?
 Every number is a multiple of itself
 Every multiple of a number is greater than or equal to that number.
 Number of multiples of a given number is infinite
Other Numbers
Prime Numbers
The numbers having exactly two factors 1 and the number itself.
Ex: 2,3,5,7,11, etc..
Composite Numbers:
The numbers having more than two factors.
Ex: 4,6,8,9,10, etc..
Even numbers:
A number which is a multiple of 2.
Ex: 2,4,6,8,10 ....
Odd Numbers:
A number which is not a multiple of 2.
Ex: 1,3,5,7,9...
Divisibility of Numbers
A number is divisible by 10 if it has 0 in its units place.
A number is divisible by 5 if it has either 0 or 5 in its units place.
A number is divisible by 2 if it has any of the digits 0, 2, 4, 6 or 8 in its units place.
A number is divisible by 3 if the sum of its digits is a multiple of 3.
A number is divisible by 6 if it is divisible by 2 and 3 both.
A number with 3 or more digits is divisible by 4 if the number formed by its last two digits (i.e., units and tens) is divisible by 4.
A number with 4 or more digits is divisible by 8 if the number formed by its last three digits is divisible by 8.
A number is divisible by 9 if the sum of all the digits of the number is divisible by 9.
A number is divisible by 11 if the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) of the number is either 0 or divisible by 11.
Common Factors
The factors which are the factors of each of the given numbers.
Ex:
 Factors of 20 are 1, 2, 4, 5, 10 and 20.
 Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
 Common Factors of 20 and 48: 1,2,4
Common Multiples
The
multiples which are the multiples of each of the given numbers.
Multiples of 2 are 2, 4, 6, 8, 10, 12
Multiples of 3 are 3, 6, 9, 12, 15, 18
Common multiples of 2 and 3 = 2,3,6,12
Factorisation and Prime Factorisation
Factorisation is expressing the number as a product of its factors.
Ex: 36 = 3 x 12 = 4 x 9
Prime Factorisation expressing the number as a product of its prime factors.
Ex: 36 = 2 x 2 x 3 x 3
HCF and LCM
The Highest Common Factor (HCF) of two or more given numbers is the highest of
their common factors. It is also known as Greatest Common Divisor (GCD).
Eg:
Find the HCF of 8 and 12?
Prime Factorisation of the numbers
8 = 2 × 2 × 2
12 = 2 × 2 × 3
Common factors are 2,2
So HCF = 2 × 2 = 4
The Lowest Common Multiple (LCM) of two or more given numbers is the lowest of their common multiples.
Eg:
Find the LCM of 8 and 12?
Prime Factorisation of the numbers
8 = 2 × 2 × 2
12 = 2 × 2 × 3
So, LCM = (2 × 2 × 2) × (3) = 24
NCERT SOLUTIONS
EXERCISE – 3.1
1. Write all the factors of the following numbers:
(a) 24
Ans:
Factors of 24 are:
24 = 1 x 24;
24 = 2 x 12;
24 = 3 x 8;
24 = 4 x 6
Hence, all the factors of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24.
(b) 15
Ans:
15 = 1 x 15;
15 = 3 x 5
Hence, all the factors of 15 are: 1, 3, 5 and 15.
(c) 21
Ans:
21 = 1 x 21;
21 = 3 x 7
Hence, all the factors of 21 are: 1, 3, 7 and 21.
(d) 27
Ans:
27 = 1 x 27;
27 = 3 x 9.
Hence, all the factors of 27 are: 1, 3, 9 and 27.
(e) 12
Ans:
12 = 1 x 12;
12 = 2 x 6;
12 = 3 x 4
Hence, all the factors of 12 are: 1, 2, 3, 4, 6 and 12.
(f) 20
Ans:
20 = 1 x 20;
20 = 2 x 10;
20 = 4 x 5
Hence, all the factors of 20 are: 1, 2, 4, 5, 10 and 20.
(g) 18
Ans:
18 = 1 x 18;
18 = 2 x 9;
18 = 3 x 6
Hence, all the factors of 18 are: 1, 2, 3, 6, 9 and 18.
(h) 23
Ans:
23 = 1 x 23
Hence, all the factors of prime number 23 are: 1 and 23.
(i) 36
Ans:
36 = 1 x 36;
36 = 2 x 18;
36 = 3 x 12;
36 = 4 x 9;
36 = 6 x 6
Hence, all the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18 and 36.
2. Write first five multiples of:
(a) 5
Ans:
First five multiples of 5 are:
5 x 1 = 5;
5 x 2 = 10;
5 x 3 = 15;
5 x 4 = 20;
5 x 5 = 25
Hence, the required multiples of 5 are: 5, 10, 15, 20 and 25.
(b) 8
Ans:
First five multiples of 8 are:
8 x 1 = 8;
8 x 2 = 16;
8 x 3 = 24;
8×4 = 32;
8 x 5 = 40
Hence, the required multiples of 8 are: 8, 16, 24, 32 and 40.
(c) 9
Ans:
First five multiples of 9 are:
9 x 1 = 9;
9 x 2 = 18;
9 x 3 = 27;
9 x 4 = 36;
9 x 5 = 45
Hence, the required multiples of 9 are: 9,18, 27, 36 and 45.
3. Match the items in column 1 with the items in column 2.
Column 1 Column 2
(i) 35 (a) Multiple of 8
(ii) 15 (b) Multiple of 7
(iii) 16 (c) Multiple of 70
(iv) 20 (d) Factor of 30
(v) 25 (e) Factor of 50
(f) Factor of 20
Ans:
(i)  (b) (7 x 5 = 35)
(ii)  (d) (15 x 2 = 30)
(iii)  (a) (8 x 2 = 16)
(iv)  (f) (20 x 1 = 20)
(v)  (e) (25 x 2 = 50)
4. Find all the multiples of 9 upto 100.
Ans:
9 x 1 = 9; 9 x 2 = 18; 9 x 3 = 27; 9 x 4 = 36;
9 x 5 = 45; 9 x 6 = 54; 9 x 7 = 63; 9 x 8 = 72;
9 x 9 = 81; 9 x 10 = 90; 9 x 11 = 99
Hence, all the multiples of 9 up to 100 are:
9, 18, 27, 36, 45, 54, 63, 72, 81, 90 and 99.
EXERCISE – 3.2
1. What is the sum of any two (a) Odd
numbers? (b) Even numbers
Ans:
The sum of any two odd numbers is even. (3 + 3 = 6)
The sum of any two even numbers is even. (2 + 2 = 4)
2. State whether the following statements are True or False:
(a) The sum of three odd numbers is even.
(b) The sum of two odd numbers and one even number is even.
(c) The product of three odd numbers is odd.
(d) If an even number is divided by 2, the quotient is always odd.
(e) All prime numbers are odd
(f) Prime numbers do not have any factors.
(g) Sum of two prime numbers is always even.
(h) 2 is the only even prime number.
(i) All even numbers are composite numbers.
(j) The product of two even numbers is always even.
Ans:
(а) False (3 + 5 + 7 = 15 (odd) )
(b) True [3 + 5 + 6 = 14 (even)]
(c) True [5 x 7 x 9 = 315 (odd)]
(d) False [6 + 2 = 3 (odd)]
(e) False [2 is a prime number but it is even]
(f) False [3 is a prime number having 1 and 3 as its factors]
(g) False [7 + 2 = 9 (odd)]
(h) True [2 is even and the lowest prime number]
(i) False [2 is even but not composite number]
(j) True [4 x 6 = 24 (even)]
3. The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers upto 100.
Ans:
The required pair of prime numbers having same digits are:
17 and 71
37 and 73
79 and 97
4. Write down separately the prime and composite numbers less than 20.
Ans:
Prime numbers less than 20 are:
2, 3, 5, 7, 11, 13, 17 and 19
Composite numbers less than 20 are:
4, 6, 8, 9, 10, 12, 14, 15, 16 and 18
5. What is the greatest prime number between 1 and 10?
Ans:
The greatest prime number between 1 and 10 is 7.
6. Express the following as the sum of two odd primes.
(a) 44 (b) 36 (c) 24 (d) 18
Ans:
(a) 44 = 13 + 31
(b) 36 = 17 + 19
(c) 24 = 7 + 17
(d) 18 = 7 + 11
7. Give three pairs of prime numbers whose difference is 2. [Remark: Two prime numbers whose difference is 2 are called twin primes].
Ans:
The three pairs of prime numbers whose difference is 2 are
3, 5
5, 7
11, 13
8. Which of the following numbers are prime?
(a) 23 (b) 51 (c) 37 (d) 26
Ans:
(a) 23 is a prime number [23 = 1 x 23]
(b) 51 is not a prime number [51 = 1 x 3 x 17]
(c) 37 is a prime number [37 = 1 x 37]
(d) 26 is not a prime number [26 = 1 x 2 x 13]
9. Write seven consecutive composite numbers less than 100 so that there is no prime number between them.
Ans:
Required seven consecutive composite numbers are:
90, 91, 92, 93, 94, 95 and 96
10. Express each of the following numbers as the sum of three odd primes:
(a) 21 (b) 31 (c) 53 (d) 61
Ans:
(a) 21 can be expressed as 3 + 5 + 13
(b) 31 can be expressed as 5 + 7 + 19
(c) 53 can be expressed as 13 + 17 + 23
(d) 61 can be expressed as 11 + 13 + 37
11. Write five pairs of prime numbers less than 20 whose sum is divisible by 5. (Hint: 3 + 7 = 10)
Ans:
Required pairs of prime numbers less than 20 are:
(i) 2 + 3 = 5
(ii) 2 + 13 = 15
(iii) 11 + 9 = 20
(iv) 17 + 3 = 20
(v) 7 + 13 = 20
12. Fill in the blanks:
(a) A number which has only two factors is called a ______.
(b) A number which has more than two factors is called a ______.
(c) 1 is neither ______ nor ______.
(d) The smallest prime number is ______.
(e) The smallest composite number is _____.
(f) The smallest even number is ______.
Ans:
(a) prime number
(b) composite number
(c) prime, composite
(d) 2
(e) 4
(f) 2
EXERCISE 3
1. Using divisibility tests, determine
which of the following numbers are divisible by 2; by 3; by 4; by 5; by 6; by
8; by 9; by 10; by 11 (say, yes or no):
Number 
Divisible by 

2 
3 
4 
5 
6 
8 
9 
10 
11 

128 
Yes 
No 
Yes 
No 
No 
Yes 
No 
No 
No 
990 
 
 
 
 
 
 
 
 
 
1586 
 
 
 
 
 
 
 
 
 
275 
 
 
 
 
 
 
 
 
 
6686 
 
 
 
 
 
 
 
 
 
639210 
 
 
 
 
 
 
 
 
 
429714 
 
 
 
 
 
 
 
 
 
2856 
 
 
 
 
 
 
 
 
 
3060 
 
 
 
 
 
 
 
 
 
406839 
 
 
 
 
 
 
 
 
 
Ans:
Number 
Divisible by 

2 
3 
4 
5 
6 
8 
9 
10 
11 

128 
Yes 
No 
Yes 
No 
No 
Yes 
No 
No 
No 
990 
Yes 
Yes 
No 
Yes 
Yes 
No 
Yes 
Yes 
Yes 
1586 
Yes 
No 
No 
No 
No 
No 
No 
No 
No 
275 
No 
No 
No 
Yes 
No 
No 
No 
No 
Yes 
6686 
Yes 
No 
No 
No 
No 
No 
No 
No 
No 
639210 
Yes 
Yes 
No 
Yes 
Yes 
No 
No 
Yes 
Yes 
429714 
Yes 
Yes 
No 
No 
Yes 
No 
Yes 
No 
No 
2856 
Yes 
Yes 
Yes 
No 
Yes 
Yes 
No 
No 
No 
3060 
Yes 
Yes 
Yes 
Yes 
Yes 
No 
Yes 
Yes 
No 
406839 
No 
Yes 
No 
No 
No 
No 
No 
No 
No 
2. Use divisibility test, to determine which number is divisible by 4 and 8
Hint:
Divisibility by 4 and 8 is checked by the last 2 and 3 digits respectively.
If the last two digit is divisible by 4, then the number is divisible by 4.
If the last three digits are divisible by 8,then the number is divisible by 8.
(a) 572
Given Last two digits are 72, it's divisible by 4. Hence 572 is also divisible by 4.
Give last three digits are 572, it's not divisible by 8, hence 572 is not divisible by 8
(b) 726352
Given Last two digits are 52, it's divisible by 4. Hence 726352 is also divisible by 4.
Give last three digits are 352, it's divisible by 8, hence 726352 is also divisible by 8.
(c) 5500
Given Last two digits are 00, Hence 5500 is also divisible by 4.
Give last three digits are 500, it's not divisible by 8, hence 5500 is not divisible by 8
(d) 6000
Given Last two digits are 00, Hence 6000 is also divisible by 4.
Give last three digits are 000, Hence 6000 is also divisible by 8.
(e) 12159
Given Last two digits are 59, it's divisible by 4. Hence 12159 is also divisible by 4.
Give last three digits are 159, it's divisible by 8, hence 12159 is also divisible by 8.
(f) 14560
Given Last two digits are 60, it's divisible by 4. Hence 14560 is also divisible by 4.
Give last three digits are 560, it's divisible by 8, hence 14560 is also divisible by 8.
EXERCISE 4
EXERCISE – 3.4
1. Find the common factors of:
(a) 20 and 28
Factors of 20 are 1, 2, 4, 5, 10, 20
Factors of 28 are 1, 2, 4, 7, 28
Hence, the common factors are 1, 2 and 4.
(b) 15 and 25
Factors of 15 are 1, 3, 5, 15
Factors of 25 are 1, 5, 25
Hence, the common factors are 1 and 5.
(c) 35 and 50
Factors of 35 are: 1, 5, 7, 35
Factors of 50 are: 1, 2, 5, 10, 50
Hence, the common factors are 1 and 5.
(d) 56 and 120
Factors of 56 are 1, 2, 4, 7, 8, 14, 28, 56
Factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, 60, 120
Hence, the common factors are 1,2, 4, and 8.
2. Find the common factors of:
(a) 4, 8 and 12
Factors of 4 are 1, 2, 4
Factors of 8 are 1, 2, 4, 8
Factors of 12 are 1, 2, 3, 4, 6, 12
Hence, the common factors are 1, 2 and 4.
(b) 5, 15 and 25
Factors of 5 are 1, 5
Factors of 15 are 1, 3, 5, 15
Factors of 25 are 1, 5, 25
Hence, the common factors are 1 and 5.
3. Find first three common multiples of:
(a) 6 and 8
First three multiples of 6 are
6 x 1= 6; 6 x 2 = 12; 6 x 3 = 18.
First three multiples of 8 are
8 x 1 = 8; 8 x 2 = 16; 8 x 3 = 24.
(b) 12 and 18
First three multiples of 12 are
12 x 1 = 12;
12 x 2 = 24;
12 x 1 = 36;
First three multiples of 18 are
18 x 1 = 18;
18 x 2 = 36;
18 x 3 = 54.
4. Write all the numbers less than 100 which are common multiples of 3 and 4.
Multiple of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99
Multiple of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100
Therefore, Common multiples of 3 and 4 = 12, 24, 36, 48, 60, 72, 84, 96
5. Which of the following numbers are coprime?
(a) 18 and 35
Factors of 18 are 1, 2, 3, 6, 9, 18
Factors of 35 are 1, 5, 7, 35
Since, the common factors of 18 and 35 is only 1.
Hence, 18 and 35 are coprime.
(b) 15 and 37
Factors of 15 are 1, 3, 5, 15
Factors of 37 are 1,37
Since, the common factor of 15 and 37 is only 1.
Hence, they are coprime.
(c) 30 and 415
Factors of 30 are 1, 2, 3, 5, 6, 15, 30
Factors of 415 are 1, 5, 83
Since, the numbers have common factors 1 and 5
Hence, they are not coprime.
(d) 17 and 68
Factors of 17 are 1, 17
Factors of 68 are 1, 2, 4, 17, 34, 68
Since, the numbers have common factors 1 and 17
Hence, they are not coprime.
(e) 216 and 215
Factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 54, 72, 108, 216
Factors of 215 are 1, 5, 43
Since only 1 is the common factor of 216 and 215.
Hence, they are coprime.
(f) 81 and 16
Factors of 81 are 1, 3, 9, 27, 81
Factors of 16 are 1, 2, 4, 8, 16
Since only 1 is common to 81 and 16
Hence, they are coprime.
6. A number is divisible by both 5 and 12. By which other number will that number be always divisible?
If the number is divisible by both 5 and 12
∴ The number will also be divisible by 5 x 12 i.e., 60.
7. A number is divisible by 12. By what other numbers will that number be divisible?
Factors of 12 are 1, 2, 3, 4, 6, 12
Hence the number which is divisible by 12, will also be divisible by its factors i.e., 1, 2, 3, 4, 6 and 12.
EXERCISE 5
EXERCISE – 3.5
1. Which of the following statements are true?
(a) If a number is divisible by 3, it must be divisible by 9.
(b) If a number is divisible by 9, it must be divisible by 3.
(c) A number is divisible by 18, if it is divisible by both 3 and 6.
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.
(e) If two numbers are coprimes, at least one of them must be prime.
(f) All numbers which are divisible by 4 must also be divisible by 8.
(g) All numbers which are divisible by 8 must also be divisible by 4.
(h) If a number exactly divides two numbers separately, it must exactly divide their sum.
(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.
Ans:
(a) False (b) True (c) False (d) True (e) False
(f) False (g) True (h) True (i) False