Practice Set 5.3
1
Find the LCM of the following numbers.
(i)
16, 24
Three Methods of Finding LCM
Select any method below to view the complete solution.
Solution :
We have to find the LCM of 16 and 24.
Listing method :
The multiples of 16 are
16, 32, 48,
64, 80, 96,
112, 128, ...
The multiples of 24 are
24, 48,
72, 96,
120, 144, ...
Common multiples of 16 and 24 are
48, 96,.... etc
The smallest common multiple of the above numbers is
48
∴ The LCM of the numbers 16 and 24 is 48
Solution :
We have to find the LCM of 16 and 24.
Prime factorisation method :
16
=
2 × 8 = 2 × 2 × 4
=
2 × 2 × 2 × 2
24
=
2 × 12 = 2 × 2 × 6
=
2 × 2 × 2 × 3
Common prime factors of 16 and 24 =
2 × 2 × 2
Uncommon factors =
2 × 3
LCM = 2 × 2 × 2 × 2 × 3 = 48
∴ LCM = 48.
Solution :
We have to find the LCM of 16 and 24.
| 2 | 16 | 24 |
| 2 | 8 | 12 |
| 2 | 4 | 6 |
| 2 | 2 | 3 |
| 3 | 1 | 3 |
| 1 | 1 |
LCM = 2 × 2 × 2 × 2 × 3
= 48
= 48
(ii)
30, 40
Three Methods of Finding LCM
Select any method below to view the complete solution.
Solution :
We have to find the LCM of 30 and 40.
Listing method :
The multiples of 30 are
30, 60, 90, 120,
150, 180, 210, 240, 270, 300, 330,
360, 390, ...
The multiples of 40 are
40, 80, 120,
160, 200, 240,
280, 320, 360,
400, ...
Common multiples of 30 and 40 are
120, 240, 360, ...
The smallest common multiple of the above numbers is
120
∴ The LCM of the numbers 30 and 40 is 120
Solution :
We have to find the LCM of 30 and 40.
Prime factorisation method :
30
=
2 × 15 = 2 × 3 × 5
40
=
2 × 20
=
2 × 2 × 10
=
2 × 2 × 2 × 5
Common prime factors =
2 × 5
Uncommon prime factors =
2 × 2 × 3
LCM = 2 × 2 × 2 × 3 × 5 = 120
Ans. The LCM of numbers 30 and 40 is 120.
Solution :
We have to find the LCM of 30 and 40.
| 2 | 30 | 40 |
| 2 | 15 | 20 |
| 2 | 15 | 10 |
| 3 | 15 | 5 |
| 5 | 5 | 5 |
| 1 | 1 |
LCM = 2 × 2 × 2 × 3 × 5
= 120
= 120
(iii)
72, 96
Three Methods of Finding LCM
Select any method below to view the complete solution.
Solution :
We have to find the LCM of 72 and 96.
Listing method :
The multiples of 72 are
72, 144, 216, 288,
360, 432, 504, 576,
648, 720, ...
The multiples of 96 are
96, 192, 288,
384, 480, 576,
672, 768.
Common multiples of 72 and 96 are
288, 576,.... etc
The smallest common multiple of the above numbers is
288
∴ The LCM of the numbers 72 and 96 is 288
Solution :
We have to find the LCM of 72 and 96.
Prime factorisation method :
72
=
2 × 36
=
2 × 2 × 18
=
2 × 2 × 2 × 9
=
2 × 2 × 2 × 3 × 3
96
=
2 × 48
=
2 × 2 × 24
=
2 × 2 × 2 × 12
=
2 × 2 × 2 × 2 × 6
=
2 × 2 × 2 × 2 × 2 × 3
Common prime factors =
2 × 2 × 2 × 3
Uncommon prime factors =
2 × 2 × 3
LCM = 2 × 2 × 2 × 2 × 2 × 3 × 3 =
288
Ans. The LCM of 72 and 96 is 288.
Solution :
We have to find the LCM of 72 and 96.
| 2 | 72 | 96 |
| 2 | 36 | 48 |
| 2 | 18 | 24 |
| 2 | 9 | 12 |
| 2 | 9 | 6 |
| 3 | 9 | 3 |
| 3 | 3 | 1 |
| 1 | 1 |
LCM = 2 × 2 × 2 × 2 × 2 × 3 × 3
= 288
= 288
(iv)
9, 15, 20
Three Methods of Finding LCM
Select any method below to view the complete solution.
Solution :
We have to find the LCM of 9, 15 and 20.
Listing method :
The multiples of 9 are
9, 18, 27, .... 180, 189, ....
360, 369, ...
The multiples of 15 are
15, 30, 45, 60, .... 180,
195, .... 360, ...
The multiples of 20 are
20, 40, .... 180, 200, .... 360, ...
Common multiples of 9, 15 and 20 are
180, 360,.... etc
The smallest common multiple of the above numbers is
180
∴ The LCM of the numbers 9, 15 and 20 is 180
Solution :
We have to find the LCM of 9, 15 and 20.
Prime factorisation method :
9
=
3 × 3
15
=
3 × 5
20
=
2 × 10
=
2 × 2 × 5
Common factors × Uncommon factors
= 2 × 2 × 3 × 3 × 5
LCM = 180
The LCM of numbers 9, 15 and 20 is 180.
Solution :
We have to find the LCM of 9, 15 and 20.
| 2 | 9 | 15 | 20 |
| 2 | 9 | 15 | 10 |
| 3 | 9 | 15 | 5 |
| 3 | 3 | 5 | 5 |
| 5 | 1 | 5 | 5 |
| 1 | 1 | 1 |
LCM = 2 × 2 × 3 × 3 × 5
= 180
= 180
(v)
30, 45, 60
Three Methods of Finding LCM
Select any method below to view the complete solution.
Solution :
We have to find the LCM of 30, 45 and 60.
Listing method :
The multiples of 30 are
30, 60, 90, 120, 150,
180, ...,
360, 390, ...
The multiples of 45 are
45, 90, 135,
180,
225, 270, ...,
360, 405, ...
The multiples of 60 are
60, 120,
180,
240, 300,
360, ...
Common multiples of 30, 45 and 60 are
180, 360,.... etc
The smallest common multiple of the above numbers is
180
∴ The LCM of the numbers 30, 45 and 60 is 180
Solution :
We have to find the LCM of 30, 45 and 60.
Prime factorisation method :
30
=
2 × 15 = 2 × 3 × 5
45
=
3 × 15 = 3 × 3 × 5
60
=
2 × 30
=
2 × 2 × 15
=
2 × 2 × 3 × 5
Common prime factors =
3 × 5
Uncommon prime factors =
2 × 2 × 3
LCM = 2 × 2 × 3 × 3 × 5 =
180
Ans. The LCM of 30, 45 and 60 is 180.
Solution :
We have to find the LCM of 30, 45 and 60.
| 2 | 30 | 45 | 60 |
| 2 | 15 | 45 | 30 |
| 3 | 15 | 45 | 15 |
| 3 | 5 | 15 | 5 |
| 5 | 5 | 5 | 5 |
| 1 | 1 | 1 |
LCM = 2 × 2 × 3 × 3 × 5
= 180
= 180
2
Solve the following examples.
(i)
Find the smallest three-digit number that is divisible by
5, 10 and 15.
Solution :
To find the lowest three-digit number divisible by
5, 10 and 15, we find their LCM.
Listing method :
Multiples of 5 :
5, 10, 15, 20, 25, 30, 35, 40, 45, 50,
55, 60, 65, 70, 75, 80, 85, 90, 95, 100,
105, 110, 115, 120, ...
Multiples of 10 :
10, 20, 30, 40, 50, 60, 70, 80, 90, 100,
110, 120, ...
Multiples of 15 :
15, 30, 45, 60, 75, 90, 105, 120, ...
Common multiples of 5, 10 and 15 :
30, 60, 90, 120, ...
The least common multiple of 5, 10 and 15 :
30
Note :
The least common multiple is 30, but the question asks
for the smallest three-digit number.
Therefore, we take the first common multiple having
three digits, which is 120.
Ans. The lowest three-digit number divisible by
5, 10 and 15 is 120.
(ii)
The product of two numbers is 360 and their GCD is 6,
then what is their LCM?
Solution :
Product of two numbers = GCD × LCM
∴ 360 = 6 × LCM
∴ LCM = 360 ÷ 6 = 60
∴ LCM = 60
Ans. The LCM of two numbers is 60.
(iii)
The LCM of two numbers is 200 and the GCD is 10.
If one of the given number is 40, what is the other number?
Solution :
Let the second number of the two numbers be x.
Product of two numbers = GCD × LCM
∴ 40 × x = 200 × 10
∴ x = 2000 ÷ 40 = 50
∴ x = 50
∴ second number = 50
Ans. The second number is 50.
(iv)
A farmer has 250 kg of wheat and 150 kg of jowar.
Both the grains are to be filled in sacks so that their
weight will be same. What is the maximum weight of grain
that can be filled in each sack?
Solution :
To find the maximum weight per sack,
we find the GCD of 250 and 150.
Divisor method :
Divisor of 250 :
1, 2, 5, 10, 25, 50, 125, 250.
Divisor of 150 :
1, 2, 3, 5, 6, 10, 15, 25, 50, 75, 150.
Common divisor of 250 and 150 :
1, 2, 5, 10, 25, 50.
The greatest common divisor of 250 and 150 =
50
∴ GCD of 250 and 150 :
50
Ans. A maximum 50 kg of grain should be filled
in each sack.
(v)
Three friends were practicing running on a circular track.
It took them 2, 3 and 4 minutes respectively to complete
one lap. If they start running from the same place at the
same time, then after how many minutes will they meet again
at the same place for the first time?
Solution :
We will find the LCM of 2, 3 and 4.
Listing method :
Multiples of 2 :
2, 4, 6, 8, 10,
12,
14, 16, 18, 20, 22, 24, ...,
34, 36, 38, ...
Multiples of 3 :
3, 6, 9,
12,
15, 18, 21,
24,
27, 30, 33,
36, 39, ...
Multiples of 4 :
4, 8,
12,
16, 20,
24,
28, 32,
36, 40, ...
Common multiples of 2, 3 and 4 :
12, 24, 36, ...
Least common multiple :
12
∴ LCM of 2, 3 and 4 :
12.
Ans. The three friends will meet again at the
starting point after 12 minutes.
(vi)
If the children in the class are lined up 6 in each row
or 8 in each row, then no child is left, what is the
minimum possible number of children in the class?
Three Methods of Finding LCM
Select any method below to view the complete solution.
Solution :
Let us find the LCM of 6 and 8.
Listing method :
Multiples of 6 :
6, 12, 18,
24,
30, 36, 42, 48, 54, 60, 66,
72, ...
Multiples of 8 :
8, 16,
24,
32, 40, 48, 56, 64,
72,
80, 88, ...
Common multiples of 6 and 8 :
24, 48, 72, ...
Least common multiple :
24.
LCM of 6 and 8 :
24.
Ans. The minimum number of children in the class is 24.
Solution :
Let us find the LCM of 6 and 8.
LCM by prime factorisation method :
6
=
2 × 3
8
=
2 × 2 × 2
Common factors :
2
Uncommon factors :
2, 2, 3
LCM = 2 × 2 × 2 × 3 =
24
Ans. The minimum number of children in the class is 24.
(vii)
Three buses going to different places leave from the same
station every 10, 15 and 20 minutes. If all the three buses
first leave at 8 am, then at what earliest time will they
leave at the same time again?
Three Methods of Finding LCM
Select any method below to view the complete solution.
Solution :
Let us find LCM of 10, 15 and 20.
Listing method :
Multiples of 10 :
10, 20, 30, 40, 50,
60,
70, 80, 90, 100, 110,
120,
130, 140, ...
Multiples of 15 :
15, 30, 45,
60,
75, 90, 105,
120, ...
Multiples of 20 :
20, 40,
60,
80, 100,
120,
140, 160, ...
Common multiples of 10, 15 and 20 :
60, 120, ...
Least common multiple :
60
∴ LCM of 10, 15 and 20 :
60.
Ans. The three buses will leave together again
after 60 minutes.
i.e. they will leave at 9 am again.
Solution :
Let us find LCM of 10, 15 and 20.
LCM by Vertical Method :
| 2 | 10 | 15 | 20 |
| 2 | 5 | 15 | 10 |
| 3 | 5 | 15 | 5 |
| 5 | 5 | 5 | 5 |
| 1 | 1 | 1 |
LCM = 2 × 2 × 3 × 5
= 60
= 60
Ans. The three buses will leave together again
after 60 minutes.
i.e. they will leave at 9 am again.
Note :
For exercise 5.3, students can use any one method to find the LCM.