whole numbers. whole numbers: numbers 0,1,2,3,4,5 and so on natural numbers: counting numbers, whole...
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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