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**WEEK 4**

**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

- 324 is divisible by 4 because the last two digits (24) are divisible by 4.
- 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

- 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

- 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.

20

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

Examples

- 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.

- 84 2. 36 3. 40 4. 24 5. 96 6. 32
- 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.

- 30 and 42 2. 21 and 56 3. 28 and 40 4. 12 and 15 5. 15 and 18
- 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

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