INTEGERS

DEFINE THE FOLLOWING WORDS;

1. Factor

2. Multiple

3. Even

4. Odd

5. Prime

6. Square

DEFINE THE FOLLOWING WORDS;

1. Factor – A number that divides another number leaving no remainder. E.g. 5 is a factor of 20.

2. Multiple – A number in the times table of another number. E.g. 20 is a multiple of 5.

3. Even – A number that is a multiple of 2.

4. Odd – A number that is not a multiple of 2.

5. Prime – A number with exactly two factors.

6. Square – A number that is the product of a number multiplied by itself. E.g. 16 is a square number because 4 x 4 = 16.

EXAMPLES

1. Find all the factors of 24

2. Find all the factors of 30

1 242 123 84 656

Factors of 24; 1, 2, 3, 4, 6, 8, 12, 24

1 302 153 1045 66

Factors of 30; 1, 2, 3, 5, 6, 10, 15, 30

YOU TRY…

1. Find all the factors of 18

2. Find all the factors of 20

3. Find all the factors of 36

4. Find all the factors of 40

5. Find all the factors of 60

6. What sort of number has an odd number of factors?

Can you fit all the numbers from 1 to 9 intothis grid so that all the definitions are true?

Factor of 24Multiple of

3Odd

number

Square Number

Prime Number

Greater than 5

TRUE OR FALSE…?

5 is a multiple of 15.5 is a factor of 15.15 is a prime number20 is a factor 10012 is a multiple of 126 is a square number21 is a factor of 218 is a multiple of 169 is a factor of 187 is a multiple of 21

RACE TIME…!

List the prime numbers in order starting from 2…

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,

47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,

101, 103, 107, 109, 113, 127, 131, 137, 139,

149, 151, 157, 163, 167, 173, 179, 181, 191,

193, 197, 199,

211, 223, 227, 229, 233, 239, 241, 251, 257,

263, 269, 271, 277, 281, 283, 293.

HIGHEST COMMON FACTOR (HCF)

Factors of 28 are 1, 2, 4, 7, 14 and 28.

Factors of 21 are 1, 3, 7 and 21.

What is the HCF of 21 and 28?

Factors of 12 are 1, 2, 3, 4, 6 and 12.

What is the HCF of 28 and 12?

What is the HCF of 12 and 21?

LOWEST COMMON MULTIPLE (LCM)

What is the lowest common multiple of 8 and 12?

8; 8, 16, 24, 32, 40…

12; 12, 24, 36, 48, 60…

What is the LCM of 12 and 20?

What is the LCM of 8 and 20?

CALCULATE THE FOLLOWING

1. The HCF of 18 and 30

2. The LCM of 18 and 30

3. The HCF of 20 and 36

4. The LCM of 20 and 36

5. The HCF of 40 and 60

6. The LCM of 40 and 60

7. The HCF of 9 and 27

8. The LCM of 9 and 27

9. The HCF of 11 and 8

10. The LCM of 11 and 8

11. The HCF of 20 and 20

12. The LCM of 20 and 20

13. The HCF of 8a2 and 6a3

14. The LCM of 8a2 and 6a3

What properties of HCF & LCM can you define looking at the questions above?

Can you find a link between a number’s HCF and LCM and the two numbers?

Basic Sudoku rules apply butwith some new added extras..!The puzzle can be solved withthe help of the numbers in thetop parts of certain cells. Thesenumbers are the products ofthe digits in all the cellshorizontally and verticallyadjacent to the cell but notincluding that cell.

Example. The cell in the top left corner of this Sudoku contains the number 20. 20 is theproduct of the digits in the two adjacent cells, which therefore must contain the digits 4 and5. The 5 cannot go in the cell below the 20 because 5 is not a factor of 96. Therefore 5 mustbe entered into the cell to the right of the cell containing 20 and 4 in the cell below.

DIVISIBILITY TESTS

When finding factors it is helpful to know a couple of tricks to test if a number is divisible by another number without a calculator.

Does anyone know any?

DIVISIBLE BY 3?

Is 42894 divisible by 3?

4+2+8+9+4=27

2+7=9

9 is a multiple of 3, so 42894 is as well!

Add digits

And again

Is 830276 divisible by 3?

8+3+0+2+7+6=25

2+5=7

7 isn’t a multiple of 3, so 830276 isn’t either

Add digits

And again

You try…a) 9362b) 103748c) 21111834

Change one thing in the ones that don’t work to make them divisible by 3.

Test

2

3

4

5

6

8

9

Even

Digits sum to a multiple of 3

Last 2 digits are divisible by 4

Ends in 5 or 0

Divisible by 3 and even

Last 3 digits are divisible by 8

Digits sum to a multiple of 9

Divisible by … ?

2 3 4 5 6 8 9

38

96

80

225

174

416

360

1. Complete the descriptions: 2. Tick or cross for divisibility by each number

Divisible by 5

14

70

51

25

98 831

110

360

What do you notice about the numbers in the parts where the circles overlap?

3. Using the divisibility tests, place each number in the Venn Diagram

24

342

45

75

65

150

14 75 12

24 98 125

25 110 30

45 150 100

51 342 256

65 360 123

70 831 225

225

125

30

100

256

123

12

Multiples of 10

Multiples of 6

Multiples of 15

Multiples of 30

INDICES

INDICES

Write

In a different way

INDICES 1. 72 or 43

2. 23 or 52

3. 45 or 83

4. 33 or 52

5. 93 or 54

6. 82 or 43

7. 24 or 42 or 33

8. 213 or 104

9. 73 or 132

10. 65 or 56

Which is bigger…?

PRIME FACTORISATION

Find a set of numbers, all of which are prime that multiply to make 72.

That is amazing! Only one way to do it!

TRY THESE ONES…

1. 40

2. 80

3. 90

4. 336

5. 1,575

Have you expressed them in index form…?

THIS IS HARD…! A METHOD FOR 40??40 40

8 5

2 4

2 2

2 205 42 22 1

Only prime numbers!

Stop when you get to 1

ON YOUR WHITEBOARDS…

40

8 5

2 4

2 2

1. 36

2. 84

YOUR TURN…

28 56 64 72

100 144

2x2x7 2x2x2x7 2x2x2x2x2x2 2x2x2x3x3

2x2x5x5 2x2x2x2x3x3

22x7 23x7 26 23x32

22x52 24x32

Extension – Using product of primes prove that 196 and 324 are square numbers. Which of the above numbers are square? Prove it using their prime factors!

23

53

2

2

1. What are the numbers in the left circle from?

2. The numbers in the right circle?3. What do you notice about the

product of the numbers in the overlap between the circles?

4. What do you notice about the product of all the numbers in the Venn Diagram?

A CHEAT…

On your whiteboards use a Venn Diagram to find the HCF and LCM of…

IN YOUR BOOKS

Using a Venn Diagram find the HCF and LCM of each of the following pairs of numbers

1. 252 and 120

2. 120 and 315

3. 360 and 315

4. 252 and 315

5. 252 and 360

1. 12 & 2520

2. 15 & 2520

3. 45 & 2520

4. 63 & 1260

5. 36 & 2520

OTHER FACTORS…

List all the factors of 40. Express all of the factors of 40 as a product of prime numbers.Can you make any other numbers using only the prime decomposition of 40?

12 = 24 = 2 x 25 = 5

8 = 2 x 2 x 210 = 2 x 520 = 2 x 2 x 540 = 2 x 2 x 2 x 5

OTHER FACTORS…

Using the fact that 60 = 2 x 2 x 3 x 5 can you calculate all the factors of 60? How about 210?

1235

2 x 2 = 42 x 3 = 6

2 x 5 = 103 x 5 = 15

2 x 2 x 3 = 122 x 2 x 5 = 202 x 3 x 5 = 30

2 x 2 x 3 x 5 = 60

COMMON FACTORS

40 = 2 x 2 x 2 x 560 = 2 x 2 x 3 x 5

Using product of primes can you write down a number which is a factor of both 40 and 60? What is the highest common factor?Using the formula calculate the lowest common multiple.

PRACTICE…

Without the Venn Diagram can you find the HCF and LCM of these pairs of numbers?

1. 48 & 280

2. 225 & 840

3. 48 & 225

4. 280 & 225

5. 840 & 280

8 & 1680

15 & 12600

3 & 3600

5 & 12600

280 & 840

QUIZ-TASTIC…!

∞ What is the smallest number with 4 different prime factors?

∞ Give two numbers, neither of which end in 0, that multiply to make 1000.

∞ What is the smallest integer you need to multiply 5445 by to make it square?

∞ The product of 3 secondary school children’s ages is 1848. How old is the eldest?

∞ What is the smallest number divisible by 1, 2, 3, 4, 5, 6 and 7?

∞ What is the smallest number that gives a remainder of 1 when divided by 2, 3, 4, 5 and 6 but is divisible by 7?