The document discusses factors and multiples of numbers. It provides examples of how Ramesh arranged 6 marbles in rows with different numbers of marbles in each row. It then explains that the numbers of marbles in each row are factors of 6. Next, it illustrates how Mary found the factors of 4 by trying different numbers as divisors. Finally, it defines multiples as numbers produced when a counting number is multiplied by another number, and provides key facts about factors and multiples.

1. Chapter 3
Playing with Numbers

2. At the end of this lesson you will be able to:
2
 Identify factors and multiples of a number
 State the properties of factors and multiples
 Classify numbers as prime, composite even and odd
 Test divisibility of numbers
 Express numbers as product of their prime factors
 Compute HCF and LCM

3. 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.
Number of rows
1 marble in each row
Total number of marbles
= 6
= 1 x 6 = 6
2 marbles in each row
Number of rows = 3
Total number of marbles
= 2 x 3 = 6

4. 4
Number of rows
3 marbles in each row
Total number of marbles
= 2
= 3 x 2 = 6
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
From these calculations, Ramesh observes that 6 can be written as a product of
two numbers in different ways such as
6 = 6 x 1 6 = 2 x 3 6 = 3 x 2 6 = 1 x 6
Number of rows
Total number of marbles
= 1
= 1 x 6 = 6

5. 5
6 x 1
2 x 3
3 x 2
1 x 6
The numbers on the left refer to the
rows and the number on the right
refer to the number of marbles in
each row
The numbers 6, 2 , 3 and 1
divide 6 exactly and leave
no remainder and
therefore they are factors
of 6.
Factors of a number are divisors that divide the number
exactly.

6. 6
Finding the factors of a number by
division.
Mary wants to find those numbers which exactly divide 4.She divides 4 by
numbers less than 4 this way.
1) 4 ( 4
- 4
0
Quotient is 4
Remainder is 0
4 = 1 x 4
2) 4 ( 2
- 4
0
Quotient is 2
Remainder is 0
4 = 2 x 2
3) 4 ( 1
- 3
1
Quotient is 1
Remainder is 1
4) 4 ( 1
- 4
0
Quotient is 1
Remainder is 0
4 = 4 x 1
She finds that the number 4 can be written as :
4 = 1 x 4; 4 = 2 x 2; 4 = 4 x 1
and knows that the numbers 1,2 and 4 are exact divisors of 4.
These numbers are called factors of 4.
Thus, a factor of a number is an exact divisor of that number.

7. Multiples
1,2,3,4,5,6,7,8,9 are counting numbers. When we multiply these
counting numbers by any number we get its multiple.
a. Collect a number of wooden/paper strips of length
4 units each.
b. Join them end to end as shown in the following figure.
c. The length of the strip at the top is 4 = 1 x 4 units
d. The length of the strip below it is 4 + 4 = 8 units.
Also,8 = 2 x 4
e. The length of the next strip is 4 + 4 + 4 = 12 units
= 3 x 4 units
f. Continuing this way we can express the other lengths as
16 = 4 x 4 ; 20 = 5 x 4
g. We say that the numbers 4,8,12,16 20… are multiples of 4.
h. The list of multiples of 4 can be continued as 24,28,32,…
i. Each of these multiples is greater than or equal to 4.
4
8
16
20
12

8. 8
Factors and Multiples
6 x 5 = 30
Multiple
Factor Factor
We can say that
the number is a
multiple of each
of its factors.

9. Important Facts about Factors and Multiples:
 1 is a factor of every number.
 Every number is a factor of itself.
 Every factor of a number is an exact divisor of that number.
 Every factor is less than or equal to the given number.
 Number of factors of a given number are finite.
 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.

10. Perfect Number
A number for which sum of
all its factors (excluding
the number itself) is equal
to the number is called a
perfect number.
Perfect
Number
Sum of its Divisors
6 1 + 2 + 3
28 1 +2 + 4 + 7 + 14
496 1 + 2 + 4 + 8 + 16 + 31 + 62 +124
+ 248
8,128 1 + 2 + 4 +8 + 16 + 32 + 64 + 127
+ 254 + 508 + 1,016 + 2,032 +
4,064