WEEK 4

SUBJECT: MATHEMATICS

TERM: 1^ST TERM

CLASS PRIMARY 6

TOPIC: HCF and LCM

 Find common factors of 2-digit whole numbers

 Find the HCF of 2-digit whole numbers

 Find common multiples of 2-digit whole numbers  find the LCM of 2-digit whole numbers.

Common factors of 2-digit whole numbers

A factor of a given number is a number that can divide the given number without a remainder. For instance 2 can divide 6 without a remainder, hence 2 is a factor of 6.

Rules for divisibility

2: A number is divisible by 2 if the last digit is an even number or zero.

3: A number is divisible by 3 if the sum of the digits is divisible by 3. For example, 4 302.

4 + 3 + 0 + 2 = 9, this is divisible by 3. Hence 4 302 is divisible by 3. Therefore, 3 is a factor of 4 302.

4: A number is divisible by 4, if the last two digits are zeros or if the last two digits of the number is divisible by 4. Examples

1. 324 is divisible by 4 because the last two digits (24) are divisible by 4.
2. 736 is divisible by 4 because the last two digits (i.e. 36) are divisible by 4. 5: A number is divisible by 5 if the last digit is either 5 or zero.

Examples

75 is divisible by 5       Hence 5 is a factor of 75

80 is divisible by 5      5 is a factor of 80

76 is not divisible by 5.     5 is not a factor of 76.

78 is divisible by 2   76 is divisible by 2

78 is also divisible by 3 but 76 is not divisible by 3.

Hence 78 is divisible by 6. (Remember 7 + 6 = 13 and 13 is not divisible by 3) ! 6 is a factor of 78. Since 2 but not 3 can divide 76 without remainder.

Thus 76 is not divisible by 6.

! 6 is not a factor of 76.

7: A number is divisible by 7 if the difference between twice the last digit and the number formed by the remaining digits is divisible by 7.

Examples

1. Consider 91 2. Consider 959

The last digit is 1. The last digit is 9.

Twice the last digit is 2 × 1 = 2. Twice the last digit is 2 × 9 = 18.

The remaining digit is 9. The remaining digits = 95.

Difference between 9 and 2 is 7. Difference between 95 and 18 = 95 – 18 = 77 Since 7 is divisible 7. Since 77 is divisible by 7.

91 is also divisible by 7. 959 is also divisible by 7.

Thus 7 is a factor of 91. Thus 7 is a factor of 959.

8: A number is divisible by 8 if the last three digits are zeros or the number is divisible by 2 without a remainder three times.

Examples

1. Consider 784 2. Consider 74

784 ÷ 2 = 392 (First division) 748 ÷ 2 = 374 (First division)

392 ÷ 2 = 196 (Second division) 374 ÷ 2 = 187 (Second division)

196 ÷ 2 = 98 (Third division) 187 ÷ 2 = 93 remainder 1 (Third division) Thus 784 ÷ 8 = 98 Here 748 cannot be divided by 2, without a remainder, three times.

Thus 8 cannot divide 748 without a remainder.

Hence 8 is factor of 784. Therefore 8 is not a factor of 748.

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9: A number is divisible by 9, if the sum of its digits is divisible by 9.

Examples

1. Consider 801 2. Consider 234

8 + 0 + 1 = 9 2 + 3 + 4 = 9

Since 9 is divisible by 9 Since 9 is divisible by 9 Then 801 is divisible by 9 Then 234 is divisible by 9 Hence 9 is a factor of 801. Hence 9 is a factor of 234.

10: A number is divisible by 10 if the last digit is ZERO. For example, 180 is divisible by 10 but 108 is not. Thus 10 is a factor of 180, but not a factor of 108.

How to find the factors of a given number

Starting from 1, find all other numbers that can divide the given number without a remainder. Examples

This method finds the factors of 48.

48 = 1 × 48

= 2 × 24

= 3 × 16

= 4 × 12

= 6 × 8

Exercise 1

For each number, list all the factors.

1. 84 2. 36 3. 40 4. 24 5. 96 6. 32
2. 80 8. 54 9. 90 10. 72 11. 19 12. 71

Common factors

Examples

Look at the method of finding the common factors of 24 and 30.

Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30

Common factors of 24 and 30 are 1, 2, 3, and 6

Exercise 2

Find the common factors of the following numbers.

1. 30 and 42 2. 21 and 56 3. 28 and 40 4. 12 and 15 5. 15 and 18
2. 25 and 75 7. 21 and 35 8. 18 and 24 9. 81 and 90 10. 24 and 60

Highest Common Factors (HCF) of 2-digit whole numbers 

Finding HCF using prime factorisation

Examples

Find the Highest Common Factor (HCF) of 18 and 30.

Solution

1. 18 = 2 × 3 × 3 2. 30 = 2 × 3 × 5

Note: The common prime factors are 2 and 3.

! Highest Common Factors (HCF) = 2 × 3 = 6

6 is the highest factor that can divide both 18 and 30.

Exercise 4

1. Find the HCF of these numbers.
2. 16, 24 and 40 2. 72, 40 and 36 3. 20, 30 and 40 4. 84, 48 and 36
3. 30, 40 and 75 6. 12, 21 and 18 7. 12, 15, and 21 8. 18, 27 and 30 B. Find the HCF of the following numbers, using the factor method.
4. 12, 24 and 48 2. 15, 25 and 35 3. 20, 25 and 40
5. 22, 33 and 44 5. 16, 20 and 24 6. 13 and 52
6. What is the highest natural number which divides exactly into 40 and 100?
7. Find the difference between the HCF of 40 and 56; and the HCF of 27 and 63.
8. The highest common factor of two numbers is 2. The larger of the two numbers is 24.

The other number has 5 as one of its factors. What is the other number?

Unit 3 Common multiples of 2-digit whole numbers

Multiples of a given number are numbers that are formed by successfully multiplying the

given number by counting numbers 1, 2, 3, 4, 5, 6 … Examples

Multiples of 3 and 5 are found here.

Multiples of 3 are 3 × 1 = 3, 3 × 2 = 6, 3 × 3 = 9, 3 × 4 = 12 …

Multiples of 3 are 3, 6, 9, 12…

Multiples of 5 are 5, 10, 15, 20…

Note: Multiples of a number do not end, so we use the sign (…) to show that there are still more.

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Common multiple: If two or more numbers have the same multiple, such a multiple is known as common multiple to the given numbers.

Examples

The common multiples of 2 and 3 are shown below.

Solution

Multiples of 2 are 2, 4, 6 , 8, 10, 12 , 14, 16, 18 , 20, 22, 24 …

Multiples of 3 are 3, 6 , 9, 12 , 15, 18 , 21, 24 , 27, 30, 33, 36,…

Common multiples are 6, 12, 18, 24…

Note: the common multiples are also multiples of 6.

Exercise 1

1. Find the common multiples of the following.
2. 15 and 10 2. 20 and 30 3. 10, 15 and 30 4. 18 and 36 5. 10 and 20 B. Write down the multiples of:
3. 2 between 11 and 17 2. 5 less than 24 3. 7 less than 30 List the common multiple of:
4. 4 and 6 less than 20 5. 5 and 7 less than 80
5. 10 and 12 less than 80 7. 11 and 12 less than 140
6. 12 and 14 less than 90 9. 12 and 15 between 20 and 130
7. 12 and 16 between 140 and 200 11. 15 and 25 between 140 and 250
8. 15 and 16 between 140 and 260 13. 18 and 27 between 100 and 200

Exercise 2

Find the common multiples of these numbers.

1. 15 and 20 2. 14 and 35 3. 15 and 30     4. 15 and 45
2. 20, 40 and 80 6. 10, 20 and 30 7. 12, 18 and 36    8. 10 and 15

 Least common multiples

Least common multiple (LCM) of two or more numbers is the least/smallest of all the common multiples of the two or more given numbers.

Least common multiple (LCM) is also known as Lowest Common Multiple.

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Examples

Method 1 (common multiples)

1. Here the LCM of 10, 15 and 30 have been found.

The multiples of 10 are: 10, 20, 30, 40, 50, and 60

The multiples of 15 are: 15, 30, 45, 60, 75, and 90

The multiples of 30 are: 30, 60, 90, 120, 150, and 180 The common multiples of 10, 15 and 30 are 30 and 60 ! The LCM of 10, 15 and 30 is 30.

2. Here we find the LCM of 9 and 12.

Solution

Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108 …

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

Common multiples of 9 and 12 are 36, 72, 108…

Least Common Multiple is the smallest/least of the three common multiples.

! LCM = 36

Exercise 1

Find the common multiples and LCM of these numbers.

1. 12 and 16 2. 12 and 24 3. 10 and 12    4. 15 and 30
2. 12 and 18 6. 12 and 15 7. 10, 20 and 30    8. 18 and 36
3. 13 and 39 10. 10 and 15 11. 15 and 20 12. 14 and 35
4. 15 and 45 14. 10, 15 and 30 15. 12, 18 and 36 16. 20, 40 and 80

Examples

Here the LCM of 9 and 12 is found using other methods.