Multiples, L.C.M. & Word Problems involving H.C.F. & L.C.M.

Multiples

Note:

The multiples of a number are

infinite.

Example 1 :

Write down all the

multiples of 4 less than 22.

Solution:

Multiples of 4 less than 22 are :

4, 8, 12, 16, 20

Example 2:

Write down the first 6 multiples of 8.

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

L.C.M.

What does it mean?

Least Common Multiple (L.C.M.)

The least common multiple

of 2 or more numbers is the

smallest multiple of the

common multiples.

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.

Method 2: Using prime factorisation

Example 1:

Find the L.C.M. of 24 and 30.

 Example 2:

Find the L.C.M. of 20, 32 and 40.

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

 Word problems involving H.C.F. and L.C.M.

Applications of H.C.F.

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.

Solution:

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?

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.

Applications of L.C.M.

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?

Solution

Practice Exercises

1. Tick in the box (True or False)

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

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.

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

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

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

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.

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.

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?

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?

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?

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/