Chapter 3: Playing with numbers

1

Multiples

• Multiples of a number can be made by multiplying the number by any whole number.
• 1X12=12
• 2X12 = 24
• 3 X12 =36
• 4X12=48

12, 24, 36, 48, 60, 72, 84, 96,108,120….are multiples of 12

       1 x 10 = 10,�      2 x 10 = 20,�      3 x 10 = 30,�      4 x 10 = 40,�      5 x 10 = 50,�      6 x 10 = 60,�      and so on ...

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2

Multiples

• 5 10 15 20 25 30 35 40 45 50 55

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3

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A clock is set to ring at every 5^th minute. Will it ring 55 minutes later?

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Multiples

• Is 12 a multiple of 3?�If you multiply 3 by 4 you get 12, so 12 is a multiple of 3.
• Is 15 a multiple of 3?�3 x 5 = 15. So 15 is a multiple of 3, (and also of 5).
• Is 21 a multiple of 6?�21 is not a multiple of 6 because you can't make 21 by multiplying 6 by any whole number.�6 x 3 = 18 and 6 x 4 = 24 but there is no whole number between 3 and 4 that could give us an answer of 21.

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What are the first five multiples of 13?

5

13 x 1 =13

13 x 2 = 26

13 x 3 = 39

13 x 4 = 52

13 x 5 = 65

13, 26, 39, 52, 65

13, 26, 39, 52, 65

Find the Missing Multiples

• 6, 12, 18, ____, ____

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• ___, 6, 9, 12, ____, ____, 21

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• ___, 24, 36, 48, 60, ____

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Class 5-Multiples &Factors, LCM &HCF

6

24

30

3

15

18

12

72

Composite and Prime numbers

7

Prime Numbers

• A prime number is a positive integer that has exactly two positive integer factors, 1 and itself.
• For example, if we list the factors of 28, we have 1, 2, 4, 7, 14, and 28. That's six factors.
• If we list the factors of 29, we only have 1 and 29. That's two factors. So we say that 29 is a prime number, but 28 isn't.

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8

Composite Numbers

• A Composite Number can be divided evenly by numbers other than 1 or itself.
• Example: is 6 a Prime Number or Composite Number?
• 6 can be divided evenly by 2, or by 3, as well as by 1 or 6:
• 6 = 1 × 6�6 = 2 × 3
• So 6 is a Composite Number

Class 5-Multiples &Factors, LCM &HCF

9

Factors

• Factors are the numbers you multiply together to get a product or a factor is a number that exactly divides another number without leaving a remainder.
• 12 can be written as the product of 2 x 6 or

2 x 6 = 12

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10

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2 and 6 are the factors of 12

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12 is the multiple of its factors 2 and 6

FACTORS

The factors of 12 are:

• 1 x 12
• 2 x 6
• 3 x 4
• 4 x 3
• 6 x 2
• 12 x 1
• 12 can be divided evenly by 1, 2, 3, 4, 6 and 12:
• 1 × 12 = 12
• 2 × 6 = 12
• 3 × 4 = 12
• So 12 is a Composite Number
• The factors of 12 are 1, 2, 3, 4, 6, and 12

11

Factors

• For example, the product 24 has several factors.
• 24 = 1 x 24
• 24 = 2 x 12
• 24 = 3 x 8
• 24 = 4 x 6
• So, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24

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Class 5-Multiples &Factors, LCM &HCF

12

Divisibility rules

• A number is divisible by 2  if the last digit is 0, 2, 4, 6 or 8.
• A number is divisible by 3  if the sum of the digits is divisible by 3.
• A number is divisible by 4  if the number formed by the last two digits is divisible by 4.
• A number is divisible by 5 if the last digit is either 0 or 5.
• A number is divisible by 10 if the last digit is 0.
• A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
• A number is divisible by 9 if the sum of the digits is divisible by 9.
• A number is divisible by 6 if it is divisible by 2 AND it is divisible by 3.

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13

Writing facors

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The factors of 48 are

1 x 48

2 x 24

3 x 16

4 x 12

6 x 8

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Another way of writing factors is:

Write your first pair of factors with a reasonable space between them, then move on to the next pair until you have them all.

This way, when you get to the 6,8 pair, you can stop because 7 is not a factor and you already have 8 in your list.

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14

Factorization & Exponents

• 60=1x60  �60=2x30 �60=3x20 �60=4x15 �60=5x12 �60=6x10

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15

60

Factorization & Prime Factorization

• The factor pairs for 60 are
• 60=1x60  �60=2x30 �60=3x20 �60=4x15 �60=5x12 �60=6x10

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60

2

30

Composite number

3

10

Composite number

2

5

60 is written as the product of its prime factors

60 = 2 x2 x 3 x 5

FACTOR TREE

• 60=4x15

=2x2 x3 x5

Prime factorization through short division

The prime factors of 60 are

2x2x3x5

2, 3, and 5 are all prime numbers, so we have prime factored 60. All we have to do now is neaten our answer up a bit. It is customary to write prime factorizations in increasing order, that is with the smallest numbers first.

Prime Factorization

• Now let us try another example: 910

19

So the prime factorization of 910 is 2x5x7x13

Least common Multiple-LCM

• A cold drink truck visits Rita's neighbourhood every 4 days and Amul ice cream truck visits her neighborhood every 5 days. For the month of June on which day will both the trucks visit on the same day?

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20

Least common Multiple-LCM

• In the given problem we have to first find the multiples of both the numbers.
• Cross out the multiples that are common. (20 and 40)
• Which multiple is the least-20
• Both the trucks will visit the neighbour hood on 20 June or after 20 days.

21

Exercise -Find the LCM of 18 and 24.�

• Common multiples of 18 and 24 are 72 and 144
• The least common multiple of 18 and 24 is 72.
• LCM = 72  

Highest Common Factor-HCF

Find the HCF of 60 and 72

The common factors of 60 and 72 are 2 x 2 x 3

The HCF of 60 and 72 =

2x2x3=12

Uncommon factors

HCF by prime factorization

LCM of 60 and 72

• To find the LCM of 60 and 72 we multiply all the factors.
• 2x2 x3x5x6 = 336

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• So the HCF of 60 and 72 is 12 and
• LCM is 336

Find the HCF of 56 and 48�

• Prime Factor Tree for 56

• Prime Factor Tree for 48

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56 is all the prime numbers 2 x 2 x 2 x 7 multiplied together

48 is the prime numbers 2 x 2 x 2 x 2 x 3 multiplied together

Look for common factors in both trees

Multiply them together =

8 is the biggest number that goes into both 56 and 48

So it is the Highest Common Factor

Finding the Highest Common Factor

HCF through Prime Factorization

• The prime factors of :

56 = 2 x 2 x 2 x 7

48 = 2 x 2 x 2 x 2 x 3

The prime numbers that are common are:

2x2x 2= 8

So the HCF of 56 and 48 is 8

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HCF & LCM as a diagram

The four different ways to find the HCF

The prime factors of :

56 = 2 x 2 x 2 x 7

48 = 2 x 2 x 2 x 2 x 3

The prime numbers that are common are:

2x2x 2= 8

So the HCF of 56 and 48 is 8

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