![Page 1: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/1.jpg)
Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M.
![Page 2: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/2.jpg)
Multiples
![Page 3: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/3.jpg)
Note:
The multiples of a number are
infinite.
![Page 4: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/4.jpg)
Example 1 :
Write down all the
multiples of 4 less than 22.
![Page 5: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/5.jpg)
Solution:
Multiples of 4 less than 22 are :
4, 8, 12, 16, 20
![Page 6: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/6.jpg)
Example 2:
Write down the first 6 multiples of 8.
![Page 7: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/7.jpg)
Solution:1 x 8 = 82 x 8 = 16
3 x 8 = 24
4 x 8 = 32
5 x 8 = 40
6 x 8 = 48
Therefore the 1st 6 multiples of 8 are: 8, 16, 24, 32, 40, and 48
![Page 8: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/8.jpg)
L.C.M.
What does it mean?
![Page 9: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/9.jpg)
Least Common Multiple (L.C.M.)
The least common multiple
of 2 or more numbers is the
smallest multiple of the
common multiples.
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Method 1: Listing multiples
Example: Find the L.C.M. of 5 and 8.
Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, …
Multiples of 8 = 8, 16, 24, 32, 40, 48, …
The L.C.M. is 40.
![Page 11: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/11.jpg)
Method 2: Using prime factorisation
Example 1:
Find the L.C.M. of 24 and 30.
![Page 12: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/12.jpg)
Example 2:
Find the L.C.M. of 20, 32 and 40.
![Page 13: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/13.jpg)
Recap
So far, we have seen how to find the factors of a number,
the prime factors of a number,
the H.C.F. of 2 or 3 numbers
the multiples of a number
the L.C.M. of 2 or 3 numbers
![Page 14: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/14.jpg)
Word problems involving H.C.F. and L.C.M.
![Page 15: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/15.jpg)
Applications of H.C.F.
![Page 16: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/16.jpg)
Example 1 :
Find the greatest number which
divides 304 to leave a remainder
of 4 and which also divides 298
to leave a remainder of 4.
![Page 17: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/17.jpg)
Solution:
![Page 18: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/18.jpg)
Example 2: Mira has two pieces of ribbon of
lengths 18 cm and 24 cm
respectively.
She wants to cut both pieces into
smaller pieces of equal length
that are as long as possible.
What would be the length of each
smaller piece?
![Page 19: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/19.jpg)
This problem can be solved using
H.C.F.
Here, we are cutting the ribbon
into smaller pieces. As we want
to have pieces of equal length,
we need to find a number that
exactly divides both 18 and 24
(that is, common factors).
As we are looking to have pieces
of ribbon that are as long as
possible, we are looking for the
highest common factor.
![Page 20: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/20.jpg)
![Page 21: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/21.jpg)
Applications of L.C.M.
![Page 22: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/22.jpg)
Example
A cinema runs its movies in
two different halls 24/7. One
movie runs for 80 minutes
and the second one runs for
120 minutes. Both movies
start at 1.00 p.m. When will
the movies begin again at
the same time?
![Page 23: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/23.jpg)
Solution
![Page 24: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/24.jpg)
Practice Exercises
1. Tick in the box (True or False)
![Page 25: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/25.jpg)
Practice Exercises2. Which of the following are
multiples of 8?
72, 98, 104, 120, 164
3. Ring all the multiples of
15 from the numbers below:
65, 45, 35, 75, 55, 95,
105
![Page 26: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/26.jpg)
Practice Exercises
4. Write down all the
multiples of 12 between 65
and 110.
5. Find the sum of all the
multiples of 7 between 50
and 80.
![Page 27: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/27.jpg)
Practice Exercises
6. (a) List all the multiples of
6 and
9 between 5 and 40.
(b) Find
(i) the common multiples
of 6 and 9
(ii) the L.C.M. of 6 and 9
![Page 28: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/28.jpg)
Practice Exercises
7. Find the L.C.M. of
(a) 15 and 20 (b) 24 and
3
(c) 48 and 72 (d) 30
and 45 (e) 18 and 54
(f) 30 and 42
![Page 29: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/29.jpg)
Practice Exercises
8. Find the L.C.M. of the
following:
(a)54, 60 and 90
(b) 34, 21 and 14
(c) 34, 119 and 196
![Page 30: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/30.jpg)
Practice Exercises
9. Find the smallest number
which is exactly divisible by 21
and 35.
10. Find the smallest number
which when divided by 15 and
24 leaves a remainder of 13.
![Page 31: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/31.jpg)
Practice Exercises
11. Find the greatest number
which will divide 3 026 and 5
053 to leave remainders 11 and
13 respectively.
12. Find the greatest number
which will divide 140 and 249
to leave remainders 8 and 18
respectively.
![Page 32: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/32.jpg)
Practice Exercises
13. Two bulbs flash at regular
intervals of 30 and 36 seconds
respectively. They flash
together at 10 45. At what time
will they flash together
(a) a second time (b) a 5th
time?
![Page 33: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/33.jpg)
Practice Exercises14. The chairs in a hall can be
arranged in rows of 48 and 60.
Find the least number of chairs
in the hall.
15. Three bells toll at intervals
of 10, 15 and 25 minutes
respectively. They start tolling
together at 7 a.m. At what time
will they next toll together?
![Page 34: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/34.jpg)
Practice Exercises16. Three bus services A, B and C
arrive at a station. Service A
arrives at the station every 15
minutes, service B arrives every
20 minutes and service C arrives
every 30 minutes. All three buses
arrive at the station together at 9
a.m. At what time will the three
buses next arrive at the station
together again?
![Page 35: Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M. 1 -Multiples, LCM and word problems.pdf · Find the L.C.M. of the following: (a)54, 60 and 90 (b) 34, 21 and 14 (c) 34,](https://reader030.vdocuments.net/reader030/viewer/2022040315/5e21428155947e280210bff8/html5/thumbnails/35.jpg)
Links https://www.calculatorsoup.com/calculators/math/
factors.php
http://www.adaptedmind.com/p.php?tagId=605
http://www.prime-numbers.net/prime-numbers-und
er-100-quiz.html
http://www.prime-numbers.net/prime-or-composit
e-1-100-worksheet.pdf
http://www.math-aids.com/Factors/
http://www.math4childrenplus.com/topics/lcm-hcf/