Prime
Numbers
Whole Numbers
What is whole numbers?
A whole number means a number that doesn’t include any fractions, negative
numbers or decimals.
Example: 0, 1, 2, 3, 4, 5, 6, …
Factors of a whole number
The factors of 20 are 1, 2, 4, 5, 10, and 20
The factors of 35 are 1, 5, 7, and 35
Classifying
Whole Numbers
Odd numbers
Whole numbers that are not
divisible by 2.
Example: 1, 3, 5, 7, 9, 11, 13, 15, …
Even numbers
Whole numbers that are
divisible by 2.
Example: 0, 2, 4, 6, 8, 10, 12, 14,
…
Composite numbers
Whole numbers that have
more than 2 different factors.
Example: 4, 6, 8, 9, 10, 12, 14, 15,
16, 18, 20, …
The product of at least two
primes.
Prime numbers
Whole numbers that have
exactly 2 different factors, 1
and itself.
Example: 2, 3, 5, 7, 11, 13, 17, 19, …
Important!
0 and 1 are neither prime nor composite.
The factor of 1 is only itself.
0 has infinite number of divisors because any nonzero whole
number divides 0 and it can’t be written as a product of two
factors.
Prime Number Test
Find the square If the number is
01 root of the number divisible by any
prime number less
03 than or equal to its
Divide the number by square root, then
all prime numbers the number is
02 less than or equal to 04 composite.
its square root.
Index
Notation
Index Notation
An expression is written as a
Index
product of a number called
the index/power/exponent
and a base. 53
Base
The base can be any number
Index Notation
5 × 5 = 52 Five squared Index
5 × 5 × 5 = 53 Five cubed
53
5 × 5 × 5 × … × 5= 5n Five to the power of n
Base
n
Index notation is useful to write number more concisely.
Example:
35 000 = 3.5 × 104
80 000 000 = 8 × 107
2 × 5 × 5 × 3 × 3 × 2 × 3 = 22 × 33 × 52
Square root
and
Cube Root
Square Root
Finding square root using prime factorisation.
Example:
Find √196 using prime factorisation.
1. Find the prime factors of 196. You can use factor trees or division by
smallest prime factor.
2 196
196 = 2 × 2 × 7 × 7
2 98
7 49
7 7
1
Square Root
2. Find the square root using one of these methods.
Method 1 Method 2
(Separate into 2 brackets) (Divide the index/indices by 2)
196 = (2 × 7) × (2 × 7) 196 = 22 × 72
196 = (2 × 7)2 √196 = 2 × 7
√196 = 2 × 7 √196 = 14
√196 = 14
Square Root
What should we do if the number is not a perfect square?
We use an estimation. See the hand out
Cube Root
Finding cube root using prime factorisation.
Example:
Find ∛216 using prime factorisation.
1. Find the prime factors of ∛216. You can use factor trees or division by
smallest prime factor.
2 216
2 108
2 54 216 = 2 × 2 × 2 × 3 × 3 × 3
3 27
3 9
3 3
1
Cube Root
2. Find the cube root using one of these methods.
Method 1 Method 2
(Separate into 3 brackets) (Divide the index/indices by 3)
196 = (2 × 3) × (2 × 3) × (2 × 3) 196 = 23 × 33
196 = (2 × 3)3 ∛196 = 2 × 3
∛196 = 2 × 3 ∛196 = 6
∛196 = 6
Cube Root
What should we do if the number is not a perfect cube?
1. Find the perfect cube that is closest to the number you want to
find the cube root for
2. Find the cube root of the closest perfect cube
3. Use approximately symbol ≈
Example:
What is the value of ∛345
The closest perfect cube to 345 is 343.
∛345 ≈ ∛343 = 7
Highest
Common
Factor
HCF
Method 1 Prime Factorisation
Find the highest common factor of 80 and 96
80 = 2 × 2 × 2 × 2 × 5 or 80 = 24 × 5
96 = 2 × 2 × 2 × 2 × 2 × 3 96 = 25 × 3
Choose the common prime
HCF = 2 × 2 × 2 × 2 = 16 HCF = 24 = 16 factors with the lowest index
Method 2 Ladder Method
Find the highest common factor of 80 and 96.
2 80 96
2 40 48 HCF = 2 × 2 × 2 × 2
2 20 24
= 16
2 10 12
5 6
Method 1 Prime Factorisation
Find the highest common factor of 90, 108, and 120
90 = 2 × 3×3 5 or 90 = 2 × 32 × 5
108 = 2 × 2 × 3×3×3 108 = 22 × 33
120 = 2 × 2 × 2 × 3 × 5 120 = 23 × 3 × 5
Choose the common prime
HCF = 2 × 3=6 HCF = 2 × 3 = 6 factors with the lowest index
Method 2 Ladder Method
Find the highest common factor of 90, 108, and 120.
2 90 108 120 HCF = 2 × 3
3 45 54 60 =6
15 18 20
Lowest
Common
Multiple
LCM
Method 1 Prime Factorisation
Find the lowest common multiple of 12 and 30
12 = 2 × 2 × 3 or 12 = 22 × 3
30 = 2 × 3×5 30 = 2 × 3 × 5
Multiply all the prime factors, for
LCM = 2 × 2 × 3 × 5 = 60 LCM = 22 × 3 × 5 = 60 common prime factors choose the
highest index.
Method 2 Ladder Method
Find the lowest common multiple of 12 and 30.
2 12 30
3 6 15 HCF = 2 × 2 × 3 × 5
2 5
= 60
Method 1 Prime Factorisation
Find the lowest common multiple of 8, 16, and 20
8 =2×2×2 or 8 = 23
16 = 2 × 2 × 2 × 2 16 = 24
20 = 2 × 2 × 5 20 = 22 × 5
LCM = 2 × 2 × 2 × 2 × 5 = 80 LCM = 24 × 5 = 80 Multiply all the prime factors, for
common prime factors choose the
highest index.
Method 2 Ladder Method
Find the lowest common multiple of 8, 16, and 20.
2 8 16 20 LCM = 2 × 2 × 2 × 2 × 5
2 4 8 10 = 80
2 2 4 5
1 2 5
