By studying this lesson, you will be able to,
● find the factors and multiples of a whole number,
● solve problems related to factors and multiples and
● examine whether a whole number is divisible by 2, 5 and 10.
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Consider a class consisting of six children. Suppose an equal number of children should sit in each row. The ways in which the 6 chairs can be arranged is shown below.
In such arrangements, the total number of 6 chairs is obtained by the product of the number of chairs in a row and the number of rows. By that it is clear that, there are many ways of writing 6, as a product of two whole numbers.
6 = 1 × 6
6 = 2 × 3
6 = 3 × 2
6 = 6 × 1
Now, let us consider how the chairs in a classroom can be arranged as above if there are twelve chairs. In this arrangement also, the total number of chairs is obtained by the product of the number of chairs in a row and the number of rows.
12 = 1 × 12
12 = 2 × 6
12 = 3 × 4
12 = 4 × 3
12 = 6 × 2
12 = 12 × 1
In this manner, any whole number can be written as a product of two whole numbers in various ways.
When a whole number is written as a product of two whole numbers, those two numbers are known as factors of the original number.
When a whole number is written as a product of two whole numbers, those two numbers are known as factors of the original number.
Since 6 = 1 × 6, the numbers 1 and 6 are factors of 6.
Since 6 = 2 × 3, the numbers 2 and 3 are also factors of 6.
When the products relevant to 6 are considered, we obtain that the factors of 6 are 1, 2, 3 and 6.
Similarly, the factors of 12 are 1, 2, 3, 4, 6 and 12.
Now, let us find the factors of 16.
We can write 16 in the following ways as a product of two whole
numbers.
16 = 1 × 16
16 = 2 × 8
16 = 4 × 4
16 = 8 × 2
16 = 16 × 1
Accordingly, the factors of 16 are 1, 2, 4, 8 and 16.
When we consider the above products relevant to 16, we see that it is
sufficient to write only the following products to obtain the factors of
16.
16 = 1 × 16 16 = 2 × 8 16 = 4 × 4
(1) Fill in the blanks with the appropriate whole numbers.
(i) 4 = 1 ×..........
4 = 2 ×..........
1, 2 and ...... are the
factors of 4.
(ii) 7 = 1 × ............
1 and .... are the factors of 7.
(iii) 8 = 1 × ..........
8 = 2 ×..........
1, 2, ...... and ...... are the
factors of 8.
(iv) 15 = 1 × 15
15 = 3 × .............
1, 3, ...... and ...... are the
factors of 15.
24 = 1 × ...........
24 = 2 ×...........
24 = 3 × .............
24 = 4 ×...........
1, 2, ......, ......, ......., ......, ...... and ...... are the factors of 24.
(vi) When the factors of 18 are written, we obtain 1, 2, ...., 6, 9 and
18.
(vii) When the factors of 40 are written, we obtain 1, 2, ...., 5, ...,10,
20 and ....
(2) Find the factors of each of the following numbers.
(i) 5 (ii) 27 (iii) 17 (iv) 22 (v) 21
(vi) 31 (vii) 32 (viii) 45 (ix) 50 (x) 60
Now let us consider how factors of a whole number are found by using
the 10 × 10 multiplication table.
Let us find the factors of 20 using the above multiplication table. For
this, let us identify the instances when 20 is obtained as the product.
20 = 2 × 10
20 = 4 × 5
The numbers 2, 4, 5 and 10 are four factors of 20.
(1) What are the factors of each of the following numbers that can be
found using the 10 ×10 multiplication table?
(i) 48 (ii) 81 (iii) 2 (iv) 28 (v) 40
(2) Obtain different ways in which 36 can be written as a product of
two whole numbers using the 10 ×10 multiplication table and fill in
the blanks.
(i) 9 × ......... (ii) 4 × .......... (iii) 6 × .........
Write down in ascending order, the factors of 36 which can be
obtained from the above products.
(3) Obtain different ways in which 9 can be written as a product of two
numbers using the 10 × 10 multiplication table and fill in the blanks.
(i)........... × ............ (ii) ............ × ..............
(4) Write down the different ways in which 30 can be written as a
product of two whole numbers using the 10 ×10 multiplication
table. Thereby write down factors of 30.
(5) Is 4 a factor of 9? Explain the reason for your answer.
When a number is divided by a factor, there is no remainder. Let us
establish this fact through the following examples. The factors of 6
have been obtained before as 1, 2, 3 and 6. When 6 is divided by each
of the numbers 1, 2, 3 and 6, there is no remainder.
6 ÷ 1 = 6 with remainder 0. 6 ÷ 2 = 3 with remainder 0.
6 ÷ 3 = 2 with remainder 0. 6 ÷ 6 = 1 with remainder 0.
Let us divide 6 by 4 and 5 which are not factors of 6.
Accordingly, when 6 is divided by any of its factors 1, 2, 3 or 6, there
is no remainder. However, when 6 is divided by 4 or 5 which are not
factors of 6, there is a remainder of 2 and 1 respectively.
If a certain whole number can be divided by another whole number
such that there is no remainder, then we identify the second number as
a factor of the first number.
Since any whole number can be divided by 1 and the number itself with
no remainder, 1 and the number itself are factors of the given number.
(1) Find three factors of each of the following numbers using the method
of division.
(i) 28 (ii) 32 (iii) 54 (iv) 90 (v) 21
(2) Is 6 a factor of 84? Explain the answer by the method of division.
(3) Is 5 a factor of 48? Explain the reason for your answer.
The answers that are obtained when 2 is multiplied by the whole
numbers 1, 2, 3, 4 and 5 are given below.
2 × 1 = 2
2 × 2 = 4
2 × 3 = 6
2 × 4 = 8
2 × 5 = 10
A number which is obtained by multiplying 2 by a whole number in this
manner is known as a multiple of 2.
In the same manner, a number which is obtained by multiplying 3 by a
whole number is known as a multiple of 3.
● 3, 6, 9, 12, 15 and 18 are several multiples of three.
Similarly,
● 5, 10, 15 and 20 are several multiples of five.
Also, observe the following results.
● All the multiples of 2 can be divided by 2 with no remainder.
● All the multiples of 3 can be divided by 3 with no remainder.
It is clear that any multiple of a whole number can be divided by that
number with no remainder.
Let us discuss more about multiples.
Consider the numbers which can be written as a product of two whole
numbers.
For example, 18 = 3 × 6.
Here 18 is obtained by multiplying 3 by 6. That is, 18 is a multiple of 3.
We may write 18 = 6 × 3 as well.
Therefore 18 is obtained by multiplying 6 by 3 and hence 18 is a multiple
of 6.
That is, 18 is a multiple of both 3 and 6.
(1) Write down five multiples of 2 which are greater than 10.
(2) Write down four multiples of 3 between 1 and 20.
(3) Write down all the multiples of 4 between 1 and 25.
(4) From the following numbers, select and write down the multiples
of three.
26, 60, 115, 48, 26, 14, 27
(5) (i) How many multiples of 9 are there between 1 and 100?
(ii) Of these numbers, which is the greatest multiple of 9?
(6) Write down three multiples of 18.
(7) What is the largest multiple of 9 which is less than 150?
(8) Write down five multiples of each of the following numbers.
(i) 4 (ii) 13 (iii) 15 (iv) 18 (v) 20
(9) Fill in the blanks.
(i) Any multiple of 10 is necessarily a multiple of ....... and .....
(ii) 11 × 7 = 77. Therefore, 77 is a multiple of .... and of .......
(10) Write down two numbers which are multiples of both 3 and 4.
(11) Write down a number which is a multiple of 2, 3 and 4.
● Solving problems related to factors and multiples
● Solving problems related to factors and multiples
Now let us solve problems related to factors and multiples.
(1) The price of a ballpoint pen is Rs 12. Find the price of 8 such pens.
Is it a multiple of 8 and 12?
(2) A water pump takes a day to fill a household tank of capacity 75
gallons.What is the number of times and the total volume of water
it can fill during a week?
(3) The price of a Rambutan is 6 rupees. Five children bought 2, 3, 4, 5
and 6 Rambutans respectively. Find the amount each student spent.
(4) A parcel containing the following items needs to be given to each
student who participated in a certain event. Find how much it will
cost to give 50 students a parcel each.
There are three gingerly rolls (thalaguli), one fish bun, two
bananas and a packet of milk in each parcel.
The price of one packet of milk is Rs 30.00
The price of a gingerly roll is Rs 5.00
The price of a fish bun is Rs 30.00
The price of a banana is Rs 10.00
(5) Separate 50 pupils into groups such that each group has an equal
number of pupils. What are the numbers which can be taken in
each group?
Through divisibility we can learn about the ability of one whole number
to divide another whole number, without a remainder.
Given two numbers, if there is no remainder when one number is
divided by the other, then the first number is said to be divisible by the
second number.
For example, when 27 is divided by 3, there is no remainder. Therefore,
27 is divisible by 3.
● Examining whether a number is divisible by 2
● Examining whether a number is divisible by 2
● Examining whether a number is divisible by 5
● Examining whether a number is divisible by 5
We have learnt earlier that multiples of 5 such as 5, 10, 15, 20, 25,
30, 35, 40, 45, ... can be divided by 5 without a remainder.
Examine the digit in the ones place of these numbers.
The digit in the ones place of these numbers is always 0 or 5.
● Examining whether a number is divisible by 10
● Examining whether a number is divisible by 10
We have learnt earlier that multiples of 10 such as 10, 20, 30, 40,
50, 60, 70, 80, 90, 100, 110, 120, ... can be divided by 10 without a
remainder.
Examine the digit in the ones place of these numbers.
The digit in the ones place of all these numbers is 0.
(1) From the following, select and write down the numbers which are
divisible by 2.
25, 33, 42, 57, 64, 69, 126, 135, 148, 250, 331, 1457, 3263, 4584,
2689, 3150, 2472
(2) What are the digits that the blank box in 128 can be filled with,
if this number consisting of four digits is divisible by 2?
(3) Select and write down the numbers which are suitable for the blank
boxes to the right of the box containing numbers. (A number could
be included in more than one box)
(4) (i) Are there numbers which are common to all three of the boxes
on the right in question 3 above? What are these numbers@
(ii) Are the numbers in box (c) essentially in boxes (a) and (b) ?
(iii) What are the numbers that are common to both boxes (a) and
(b) in question 3 above. Examine whether these numbers are
in box (c) as well. Accordingly, write down the conclusion
that you can draw.
(5) Join the points relevant to the numbers which are divisible by 2
in ascending order. Then, join the points relevant to the smallest
number and the largest number you joined.
(6) Join the points relevant to the numbers which are divisible by 5
in ascending order. Then join the points relevant to the smallest
number and the largest number you joined.
(7) Join the points relevant to the numbers which are divisible by 10
in ascending order. Then join the points relevant to the smallest
number and the largest number you joined.
(1) Give reasons why 7 is not a factor of 45.
(2) All the factors of a certain number, apart from the number itself are
1, 2, 3, 4 and 6.What is this number ?
(3) The number of marbles in a certain box is a multiple of 6. When
this number is rounded off to the nearest multiple of ten, the value
obtained is 40. Write down the two values that the number of
marbles in the box can take.
(4) The number of biscuits there are in a certain packet of biscuits is a
multiple of four which is less than 20. When this number is rounded
off to the nearest multiple of 10, the value obtained is 20. How many
biscuits are there in the packet?
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