Whole Numbers

Whole Numbers

Whole numbers: Numbers 0,1,2,3,4,5 and so on

Natural Numbers: Counting numbers, whole numbers but without zero, or positive whole numbers

Operations with Whole Numbers

To combine whole numbers, we have operations

Addition (+): 34 + 26

Subtraction (-): 18 – 14

Multiplication (x): 3 x 7

Division (÷): 30 ÷ 6

Order of Operations

BEDMAS

Brackets, Exponents, Multiplication/Division, Addition/Subtraction

Operations on the same level (x/÷, +/-) work from left to right

Within brackets, do innermost brackets first

Questions

32÷4+9

234 ÷ (10+3) – 9

2 + 14 - 3 + 5 + 7 - 25 x 0

56 – 7 x 8

[6-4 x (3 x {4-4})] + 1

Divisibility

A number is divisible by another number if when you divide them, there is no remainder

14 is divisible by 7 because it is 2 with no remainder

Divisibility Rules

2: one’s digit is even

3: sum of digits is divisible by 3

4: last 2 digits are divisible by 4

5: number ends in 0 or 5

6: divisible by both 2 and 3

9: sum of digits is divisible by 9

10: number ends in 0

Factors and Multiples

Prime numbers: a natural number that exactly have 2 factors

3, 5, 19

Factor: if a number is divisible by the number, then it is a factor of that number

7 is a factor of 21

Multiple: the product of the number and some other whole number

Multiples of 3: 3, 6, 9, 12, 15…

Composite number: a natural number that has 3 or more factors

24, 15, 100

Prime Factorization

Factor Tree:

60

Prime factorization of 60: 2x2x3x5In exponential form or index notation:

22x3x5

Prime Factorization con’t

Repeated division

60 ÷ 2

30 ÷ 2

15 ÷ 5

3 ÷ 3

1

Prime Factors

Finding Factors

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

1 x 24 = 24

2 x 12 = 24

3 x 8 = 24

4 x 6 = 24

Finding Factors con’t

Using Prime Factors: 120 = 23 x 31 x 51

x 50 30 31

20 1 3

21 2 6

22 4 12

23 8 24

x 51 30 31

20 5 15

21 10 30

22 20 60

23 40 120

Counting Factors

Rainbow method:

Factors of 24: there are 8 factors of 24

Counting Factors con’t

24 = 23 x 31

x 30 31

20 1 3

21 2 6

22 4 12

23 8 24

2 options

4 options

Take the exponent numbers and add 1. Then, multiply them all together.

(3+1) x (1+1) = 8 factors

Highest Common Factor

72, 54

72 = 2x3x3 x2x2

54 = 2x3x3

HCF(54,72) = 2x3x3 = 18

Find all common prime factors, and multiply together.

HCF con’t

÷ 72 54

2 36 27

3 12 9

3 4 3

2x3x3 = 18

HCF con’t

233

322

7254

Common Factors

Least Common Multiple

Multiples of 12: 12, 24, 36, 48, 60, 72…

Multiples of 18: 18, 36, 54, 72…

Common Multiples of 12 and 18 are 36, 72

Least common multiple is 36

LCM con’t

12 = 2 x 3 x 2

18 = 2 x 3 x 3

HCF(12,18) = 6

LCM = HCF x (product of remaining prime factors)

6 x (2 x 3) = 36

LCM(12,18)= 36

Questions

( 100÷ 5+ 6) – 7 x 2 + 18 ÷ 2

[ ( 27 + 45) ÷ 9 + 42 ÷ 6 ] x 3

Find the LCM and HCF of 45, 18

Prime factorize 1000

How many factors does 96 have?

Which of the following is divisible by 6?

45, 23, 36, 27, 96, 78