Page 1
Ramesh has 6 marbles with him. He wants to arrange them in rows in such a way
that each row has the same number of marbles. He arranges them in the following
ways and matches the total number of marbles.
(i) 1 marble in each row
Number of rows = 6
Total number of marbles = 1 × 6 = 6
(ii) 2 marbles in each row
Number of rows = 3
Total number of marbles = 2 × 3 = 6
(iii) 3 marbles in each row
Number of rows = 2
Total number of marbles = 3 × 2 = 6
(iv) He could not think of any arrangement in which each row had 4 marbles or
5 marbles. So, the only possible arrangement left was with all the 6 marbles
in a row.
Number of rows = 1
Total number of marbles = 6 × 1 = 6
From these calculations Ramesh observes that 6 can be written as a product
of two numbers in different ways as
6 = 1 × 6; 6 = 2 × 3; 6 = 3 × 2; 6 = 6 × 1
3.1 Introduction
Chapter 3 Chapter 3 Chapter 3 Chapter 3 Chapter 3
Playing with Playing with
Playing with Playing with Playing with
Numbers Numbers
Numbers Numbers Numbers
Rationalised 2023-24
Page 2
Ramesh has 6 marbles with him. He wants to arrange them in rows in such a way
that each row has the same number of marbles. He arranges them in the following
ways and matches the total number of marbles.
(i) 1 marble in each row
Number of rows = 6
Total number of marbles = 1 × 6 = 6
(ii) 2 marbles in each row
Number of rows = 3
Total number of marbles = 2 × 3 = 6
(iii) 3 marbles in each row
Number of rows = 2
Total number of marbles = 3 × 2 = 6
(iv) He could not think of any arrangement in which each row had 4 marbles or
5 marbles. So, the only possible arrangement left was with all the 6 marbles
in a row.
Number of rows = 1
Total number of marbles = 6 × 1 = 6
From these calculations Ramesh observes that 6 can be written as a product
of two numbers in different ways as
6 = 1 × 6; 6 = 2 × 3; 6 = 3 × 2; 6 = 6 × 1
3.1 Introduction
Chapter 3 Chapter 3 Chapter 3 Chapter 3 Chapter 3
Playing with Playing with
Playing with Playing with Playing with
Numbers Numbers
Numbers Numbers Numbers
Rationalised 2023-24
PLAYING WITH NUMBERS
25
From 6 = 2 × 3 it can be said that 2 and 3 exactly divide 6. So, 2 and 3 are
exact divisors of 6. From the other product 6 = 1 × 6, the exact divisors of 6 are
found to be 1 and 6.
Thus, 1, 2, 3 and 6 are exact divisors of 6. They are called the factors of 6.
Try arranging 18 marbles in rows and find the factors of 18.
3.2 Factors and Multiples
Mary wants to find those numbers which exactly divide 4. She divides 4 by
numbers less than 4 this way.
1) 4 (4 2) 4 (2 3) 4 (1 4) 4 (1
– 4 – 4 – 3 – 4
0 0 1 0
Quotient is 4 Quotient is 2 Quotient is 1 Quotient is 1
Remainder is 0 Remainder is 0 Remainder is 1 Remainder is 0
4 = 1 × 4 4 = 2 × 2 4 = 4 × 1
She finds that the number 4 can be written as: 4 = 1 × 4; 4 = 2 × 2;
4 = 4 × 1 and knows that the numbers 1, 2 and 4 are exact divisors of 4.
These numbers are called factors of 4.
A factor of a number is an exact divisor of that number.
Observe each of the factors of 4 is less than or equal to 4.
Game-1 : This is a game to be played by two persons say A and B. It is
about spotting factors.
It requires 50 pieces of cards numbered 1 to 50.
Arrange the cards on the table like this.
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31 32 33 34 35
36 37 38 39 40 41 42
43 44 45 46 47 48 49 50
Rationalised 2023-24
Page 3
Ramesh has 6 marbles with him. He wants to arrange them in rows in such a way
that each row has the same number of marbles. He arranges them in the following
ways and matches the total number of marbles.
(i) 1 marble in each row
Number of rows = 6
Total number of marbles = 1 × 6 = 6
(ii) 2 marbles in each row
Number of rows = 3
Total number of marbles = 2 × 3 = 6
(iii) 3 marbles in each row
Number of rows = 2
Total number of marbles = 3 × 2 = 6
(iv) He could not think of any arrangement in which each row had 4 marbles or
5 marbles. So, the only possible arrangement left was with all the 6 marbles
in a row.
Number of rows = 1
Total number of marbles = 6 × 1 = 6
From these calculations Ramesh observes that 6 can be written as a product
of two numbers in different ways as
6 = 1 × 6; 6 = 2 × 3; 6 = 3 × 2; 6 = 6 × 1
3.1 Introduction
Chapter 3 Chapter 3 Chapter 3 Chapter 3 Chapter 3
Playing with Playing with
Playing with Playing with Playing with
Numbers Numbers
Numbers Numbers Numbers
Rationalised 2023-24
PLAYING WITH NUMBERS
25
From 6 = 2 × 3 it can be said that 2 and 3 exactly divide 6. So, 2 and 3 are
exact divisors of 6. From the other product 6 = 1 × 6, the exact divisors of 6 are
found to be 1 and 6.
Thus, 1, 2, 3 and 6 are exact divisors of 6. They are called the factors of 6.
Try arranging 18 marbles in rows and find the factors of 18.
3.2 Factors and Multiples
Mary wants to find those numbers which exactly divide 4. She divides 4 by
numbers less than 4 this way.
1) 4 (4 2) 4 (2 3) 4 (1 4) 4 (1
– 4 – 4 – 3 – 4
0 0 1 0
Quotient is 4 Quotient is 2 Quotient is 1 Quotient is 1
Remainder is 0 Remainder is 0 Remainder is 1 Remainder is 0
4 = 1 × 4 4 = 2 × 2 4 = 4 × 1
She finds that the number 4 can be written as: 4 = 1 × 4; 4 = 2 × 2;
4 = 4 × 1 and knows that the numbers 1, 2 and 4 are exact divisors of 4.
These numbers are called factors of 4.
A factor of a number is an exact divisor of that number.
Observe each of the factors of 4 is less than or equal to 4.
Game-1 : This is a game to be played by two persons say A and B. It is
about spotting factors.
It requires 50 pieces of cards numbered 1 to 50.
Arrange the cards on the table like this.
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31 32 33 34 35
36 37 38 39 40 41 42
43 44 45 46 47 48 49 50
Rationalised 2023-24
MATHEMATICS
26
Steps
(a) Decide who plays first, A or B.
(b) Let A play first. He picks up a card from the table, and keeps it with him.
Suppose the card has number 28 on it.
(c) Player B then picks up all those cards having numbers which are factors of
the number on A ’s card (i.e. 28), and puts them in a pile near him.
(d) Player B then picks up a card from the table and keeps it with him. From the
cards that are left, A picks up all those cards whose numbers are factors of
the number on B’s card. A puts them on the previous card that he collected.
(e) The game continues like this until all the cards are used up.
(f) A will add up the numbers on the cards that he has collected. B too will do
the same with his cards. The player with greater sum will be the winner.
The game can be made more interesting by increasing the number of cards.
Play this game with your friend. Can you find some way to win the game?
When we write a number 20 as 20 = 4 × 5, we say 4
and 5 are factors of 20. We also say that 20 is a multiple
of 4 and 5.
The representation 24 = 2 × 12 shows that 2 and 12
are factors of 24, whereas 24 is a multiple of 2 and 12.
We can say that a number is a multiple of each of its
factors
Let us now see some interesting facts about factors and
multiples.
(a) Collect a number of wooden/paper strips of length 3
units each.
(b) Join them end to end as shown in the following
figure.
The length of the strip at the top is 3 = 1 × 3 units.
The length of the strip below it is 3 + 3 = 6 units.
Also, 6 = 2 × 3. The length of the next strip is 3 + 3 +
3 = 9 units, and 9 = 3 × 3. Continuing this way we
can express the other lengths as,
12 = 4 × 3 ; 15 = 5 × 3
We say that the numbers 3, 6, 9, 12, 15 are multiples of 3.
The list of multiples of 3 can be continued as 18, 21, 24, ...
Each of these multiples is greater than or equal to 3.
The multiples of the number 4 are 4, 8, 12, 16, 20, 24, ...
The list is endless. Each of these numbers is greater than or equal to 4.
multiple
?
4 × 5 = 20
? ?
factor factor
Find the possible
factors of 45, 30
and 36.
3 3
3 3 6
3 3 3 9
3 3 3 3 12
3 3 3 3 3 15
Rationalised 2023-24
Page 4
Ramesh has 6 marbles with him. He wants to arrange them in rows in such a way
that each row has the same number of marbles. He arranges them in the following
ways and matches the total number of marbles.
(i) 1 marble in each row
Number of rows = 6
Total number of marbles = 1 × 6 = 6
(ii) 2 marbles in each row
Number of rows = 3
Total number of marbles = 2 × 3 = 6
(iii) 3 marbles in each row
Number of rows = 2
Total number of marbles = 3 × 2 = 6
(iv) He could not think of any arrangement in which each row had 4 marbles or
5 marbles. So, the only possible arrangement left was with all the 6 marbles
in a row.
Number of rows = 1
Total number of marbles = 6 × 1 = 6
From these calculations Ramesh observes that 6 can be written as a product
of two numbers in different ways as
6 = 1 × 6; 6 = 2 × 3; 6 = 3 × 2; 6 = 6 × 1
3.1 Introduction
Chapter 3 Chapter 3 Chapter 3 Chapter 3 Chapter 3
Playing with Playing with
Playing with Playing with Playing with
Numbers Numbers
Numbers Numbers Numbers
Rationalised 2023-24
PLAYING WITH NUMBERS
25
From 6 = 2 × 3 it can be said that 2 and 3 exactly divide 6. So, 2 and 3 are
exact divisors of 6. From the other product 6 = 1 × 6, the exact divisors of 6 are
found to be 1 and 6.
Thus, 1, 2, 3 and 6 are exact divisors of 6. They are called the factors of 6.
Try arranging 18 marbles in rows and find the factors of 18.
3.2 Factors and Multiples
Mary wants to find those numbers which exactly divide 4. She divides 4 by
numbers less than 4 this way.
1) 4 (4 2) 4 (2 3) 4 (1 4) 4 (1
– 4 – 4 – 3 – 4
0 0 1 0
Quotient is 4 Quotient is 2 Quotient is 1 Quotient is 1
Remainder is 0 Remainder is 0 Remainder is 1 Remainder is 0
4 = 1 × 4 4 = 2 × 2 4 = 4 × 1
She finds that the number 4 can be written as: 4 = 1 × 4; 4 = 2 × 2;
4 = 4 × 1 and knows that the numbers 1, 2 and 4 are exact divisors of 4.
These numbers are called factors of 4.
A factor of a number is an exact divisor of that number.
Observe each of the factors of 4 is less than or equal to 4.
Game-1 : This is a game to be played by two persons say A and B. It is
about spotting factors.
It requires 50 pieces of cards numbered 1 to 50.
Arrange the cards on the table like this.
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31 32 33 34 35
36 37 38 39 40 41 42
43 44 45 46 47 48 49 50
Rationalised 2023-24
MATHEMATICS
26
Steps
(a) Decide who plays first, A or B.
(b) Let A play first. He picks up a card from the table, and keeps it with him.
Suppose the card has number 28 on it.
(c) Player B then picks up all those cards having numbers which are factors of
the number on A ’s card (i.e. 28), and puts them in a pile near him.
(d) Player B then picks up a card from the table and keeps it with him. From the
cards that are left, A picks up all those cards whose numbers are factors of
the number on B’s card. A puts them on the previous card that he collected.
(e) The game continues like this until all the cards are used up.
(f) A will add up the numbers on the cards that he has collected. B too will do
the same with his cards. The player with greater sum will be the winner.
The game can be made more interesting by increasing the number of cards.
Play this game with your friend. Can you find some way to win the game?
When we write a number 20 as 20 = 4 × 5, we say 4
and 5 are factors of 20. We also say that 20 is a multiple
of 4 and 5.
The representation 24 = 2 × 12 shows that 2 and 12
are factors of 24, whereas 24 is a multiple of 2 and 12.
We can say that a number is a multiple of each of its
factors
Let us now see some interesting facts about factors and
multiples.
(a) Collect a number of wooden/paper strips of length 3
units each.
(b) Join them end to end as shown in the following
figure.
The length of the strip at the top is 3 = 1 × 3 units.
The length of the strip below it is 3 + 3 = 6 units.
Also, 6 = 2 × 3. The length of the next strip is 3 + 3 +
3 = 9 units, and 9 = 3 × 3. Continuing this way we
can express the other lengths as,
12 = 4 × 3 ; 15 = 5 × 3
We say that the numbers 3, 6, 9, 12, 15 are multiples of 3.
The list of multiples of 3 can be continued as 18, 21, 24, ...
Each of these multiples is greater than or equal to 3.
The multiples of the number 4 are 4, 8, 12, 16, 20, 24, ...
The list is endless. Each of these numbers is greater than or equal to 4.
multiple
?
4 × 5 = 20
? ?
factor factor
Find the possible
factors of 45, 30
and 36.
3 3
3 3 6
3 3 3 9
3 3 3 3 12
3 3 3 3 3 15
Rationalised 2023-24
PLAYING WITH NUMBERS
27
Let us see what we conclude about factors and multiples:
1. Is there any number which occurs as a factor of every number ? Y es. It is 1.
For example 6 = 1 × 6, 18 = 1 × 18 and so on. Check it for a few more
numbers.
We say 1 is a factor of every number .
2. Can 7 be a factor of itself ? Y es. Y ou can write 7 as 7 = 7 × 1. What about 10?
and 15?.
You will find that every number can be expressed in this way.
We say that every number is a factor of itself.
3. What are the factors of 16? They are 1, 2, 4, 8, 16. Out of these factors do
you find any factor which does not divide 16? Try it for 20; 36.
You will find that every factor of a number is an exact divisor of
that number.
4. What are the factors of 34? They are 1, 2, 17 and 34 itself. Out of these
which is the greatest factor? It is 34 itself.
The other factors 1, 2 and 17 are less than 34. Try to check this for 64,
81 and 56.
We say that every factor is less than or equal to the given number.
5. The number 76 has 5 factors. How many factors does 136 or 96 have? You
will find that you are able to count the number of factors of each of these.
Even if the numbers are as large as 10576, 25642 etc. or larger, you
can still count the number of factors of such numbers, (though you may
find it difficult to factorise such numbers).
We say that number of factors of a given number are finite.
6. What are the multiples of 7? Obviously, 7, 14, 21, 28,... You will find that
each of these multiples is greater than or equal to 7. Will it happen with
each number? Check this for the multiples of 6, 9 and 10.
We find that every multiple of a number is greater than or equal to
that number.
7. Write the multiples of 5. They are 5, 10, 15, 20, ... Do you think this
list will end anywhere? No! The list is endless. Try it with multiples of
6,7 etc.
We find that the number of multiples of a given number is infinite.
8. Can 7 be a multiple of itself ? Y es, because 7 = 7×1. Will it be true for other
numbers also? Try it with 3, 12 and 16.
You will find that every number is a multiple of itself.
Rationalised 2023-24
Page 5
Ramesh has 6 marbles with him. He wants to arrange them in rows in such a way
that each row has the same number of marbles. He arranges them in the following
ways and matches the total number of marbles.
(i) 1 marble in each row
Number of rows = 6
Total number of marbles = 1 × 6 = 6
(ii) 2 marbles in each row
Number of rows = 3
Total number of marbles = 2 × 3 = 6
(iii) 3 marbles in each row
Number of rows = 2
Total number of marbles = 3 × 2 = 6
(iv) He could not think of any arrangement in which each row had 4 marbles or
5 marbles. So, the only possible arrangement left was with all the 6 marbles
in a row.
Number of rows = 1
Total number of marbles = 6 × 1 = 6
From these calculations Ramesh observes that 6 can be written as a product
of two numbers in different ways as
6 = 1 × 6; 6 = 2 × 3; 6 = 3 × 2; 6 = 6 × 1
3.1 Introduction
Chapter 3 Chapter 3 Chapter 3 Chapter 3 Chapter 3
Playing with Playing with
Playing with Playing with Playing with
Numbers Numbers
Numbers Numbers Numbers
Rationalised 2023-24
PLAYING WITH NUMBERS
25
From 6 = 2 × 3 it can be said that 2 and 3 exactly divide 6. So, 2 and 3 are
exact divisors of 6. From the other product 6 = 1 × 6, the exact divisors of 6 are
found to be 1 and 6.
Thus, 1, 2, 3 and 6 are exact divisors of 6. They are called the factors of 6.
Try arranging 18 marbles in rows and find the factors of 18.
3.2 Factors and Multiples
Mary wants to find those numbers which exactly divide 4. She divides 4 by
numbers less than 4 this way.
1) 4 (4 2) 4 (2 3) 4 (1 4) 4 (1
– 4 – 4 – 3 – 4
0 0 1 0
Quotient is 4 Quotient is 2 Quotient is 1 Quotient is 1
Remainder is 0 Remainder is 0 Remainder is 1 Remainder is 0
4 = 1 × 4 4 = 2 × 2 4 = 4 × 1
She finds that the number 4 can be written as: 4 = 1 × 4; 4 = 2 × 2;
4 = 4 × 1 and knows that the numbers 1, 2 and 4 are exact divisors of 4.
These numbers are called factors of 4.
A factor of a number is an exact divisor of that number.
Observe each of the factors of 4 is less than or equal to 4.
Game-1 : This is a game to be played by two persons say A and B. It is
about spotting factors.
It requires 50 pieces of cards numbered 1 to 50.
Arrange the cards on the table like this.
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31 32 33 34 35
36 37 38 39 40 41 42
43 44 45 46 47 48 49 50
Rationalised 2023-24
MATHEMATICS
26
Steps
(a) Decide who plays first, A or B.
(b) Let A play first. He picks up a card from the table, and keeps it with him.
Suppose the card has number 28 on it.
(c) Player B then picks up all those cards having numbers which are factors of
the number on A ’s card (i.e. 28), and puts them in a pile near him.
(d) Player B then picks up a card from the table and keeps it with him. From the
cards that are left, A picks up all those cards whose numbers are factors of
the number on B’s card. A puts them on the previous card that he collected.
(e) The game continues like this until all the cards are used up.
(f) A will add up the numbers on the cards that he has collected. B too will do
the same with his cards. The player with greater sum will be the winner.
The game can be made more interesting by increasing the number of cards.
Play this game with your friend. Can you find some way to win the game?
When we write a number 20 as 20 = 4 × 5, we say 4
and 5 are factors of 20. We also say that 20 is a multiple
of 4 and 5.
The representation 24 = 2 × 12 shows that 2 and 12
are factors of 24, whereas 24 is a multiple of 2 and 12.
We can say that a number is a multiple of each of its
factors
Let us now see some interesting facts about factors and
multiples.
(a) Collect a number of wooden/paper strips of length 3
units each.
(b) Join them end to end as shown in the following
figure.
The length of the strip at the top is 3 = 1 × 3 units.
The length of the strip below it is 3 + 3 = 6 units.
Also, 6 = 2 × 3. The length of the next strip is 3 + 3 +
3 = 9 units, and 9 = 3 × 3. Continuing this way we
can express the other lengths as,
12 = 4 × 3 ; 15 = 5 × 3
We say that the numbers 3, 6, 9, 12, 15 are multiples of 3.
The list of multiples of 3 can be continued as 18, 21, 24, ...
Each of these multiples is greater than or equal to 3.
The multiples of the number 4 are 4, 8, 12, 16, 20, 24, ...
The list is endless. Each of these numbers is greater than or equal to 4.
multiple
?
4 × 5 = 20
? ?
factor factor
Find the possible
factors of 45, 30
and 36.
3 3
3 3 6
3 3 3 9
3 3 3 3 12
3 3 3 3 3 15
Rationalised 2023-24
PLAYING WITH NUMBERS
27
Let us see what we conclude about factors and multiples:
1. Is there any number which occurs as a factor of every number ? Y es. It is 1.
For example 6 = 1 × 6, 18 = 1 × 18 and so on. Check it for a few more
numbers.
We say 1 is a factor of every number .
2. Can 7 be a factor of itself ? Y es. Y ou can write 7 as 7 = 7 × 1. What about 10?
and 15?.
You will find that every number can be expressed in this way.
We say that every number is a factor of itself.
3. What are the factors of 16? They are 1, 2, 4, 8, 16. Out of these factors do
you find any factor which does not divide 16? Try it for 20; 36.
You will find that every factor of a number is an exact divisor of
that number.
4. What are the factors of 34? They are 1, 2, 17 and 34 itself. Out of these
which is the greatest factor? It is 34 itself.
The other factors 1, 2 and 17 are less than 34. Try to check this for 64,
81 and 56.
We say that every factor is less than or equal to the given number.
5. The number 76 has 5 factors. How many factors does 136 or 96 have? You
will find that you are able to count the number of factors of each of these.
Even if the numbers are as large as 10576, 25642 etc. or larger, you
can still count the number of factors of such numbers, (though you may
find it difficult to factorise such numbers).
We say that number of factors of a given number are finite.
6. What are the multiples of 7? Obviously, 7, 14, 21, 28,... You will find that
each of these multiples is greater than or equal to 7. Will it happen with
each number? Check this for the multiples of 6, 9 and 10.
We find that every multiple of a number is greater than or equal to
that number.
7. Write the multiples of 5. They are 5, 10, 15, 20, ... Do you think this
list will end anywhere? No! The list is endless. Try it with multiples of
6,7 etc.
We find that the number of multiples of a given number is infinite.
8. Can 7 be a multiple of itself ? Y es, because 7 = 7×1. Will it be true for other
numbers also? Try it with 3, 12 and 16.
You will find that every number is a multiple of itself.
Rationalised 2023-24
MATHEMATICS
28
The factors of 6 are 1, 2, 3 and 6. Also, 1+2+3+6 = 12 = 2 × 6. We find that
the sum of the factors of 6 is twice the number 6. All the factors of 28 are 1, 2,
4, 7, 14 and 28. Adding these we have, 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × 28.
The sum of the factors of 28 is equal to twice the number 28.
A number for which sum of all its factors is equal to twice the number is
called a perfect number . The numbers 6 and 28 are perfect numbers.
Is 10 a perfect number?
Example 1 : Write all the factors of 68.
Solution : We note that
68 = 1 × 68 68 = 2 × 34
68 = 4 × 17 68 = 17 × 4
Stop here, because 4 and 17 have occurred earlier.
Thus, all the factors of 68 are 1, 2, 4, 17, 34 and 68.
Example 2 : Find the factors of 36.
Solution : 36 = 1 × 36 36 = 2 × 18 36 = 3 × 12
36 = 4 × 9 36 = 6 × 6
Stop here, because both the factors (6) are same. Thus, the factors are 1, 2,
3, 4, 6, 9, 12, 18 and 36.
Example 3 : Write first five multiples of 6.
Solution : The required multiples are: 6×1= 6, 6×2 = 12, 6×3 = 18, 6×4 = 24,
6×5 = 30 i.e. 6, 12, 18, 24 and 30.
EXERCISE 3.1
1. Write all the factors of the following numbers :
(a) 24 (b) 15 (c) 21
(d) 27 (e) 12 (f) 20
(g) 18 (h) 23 (i) 36
2. Write first five multiples of :
(a) 5 (b) 8 (c) 9
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
Rationalised 2023-24
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