PRIME FACTORS
Meaning of Prime Numbers
Prime numbers are numbers that have only two factors — 1 and itself.
In other words, a prime number can only be divided exactly by 1 and by itself.
Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, etc.
Note:
- 1 is not a prime number because it has only one factor (1).
- 2 is the smallest and the only even prime number.
A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself.
It cannot be divided evenly by any other whole number except 1 and itself.
Meaning of Composite Number
A composite number is a whole number greater than 1 that has more than two factors.
That means besides 1 and itself, there are other whole numbers that divide it exactly.
Examples: 4, 6, 8, 9, 10, 12, 14, 15, …
Definition: Prime Factor
A prime factor of a number is a factor (i.e. a divisor) of that number which is also a prime number.
Prime factors are prime numbers that multiply together to give the original number.
PRIME FACTORISATION
Meaning of Prime Factor
A prime factor is a factor of a number that is also a prime number.
For example:
The factors of 12 are 1, 2, 3, 4, 6, and 12.
The prime factors among them are 2 and 3 because they are prime numbers.
So, the prime factors of 12 are 2 and 3.
Meaning of Prime Factorisation
Prime factorisation means expressing a number as the product of its prime factors.
In other words, we break the number down until we get only prime numbers.
There are two common methods used for finding prime factors:
- Division Method
- Factor Tree Method
How to Find the Prime Factors of a Number (Step by Step)
- Start with the given number (which is a composite number, not prime).
- Begin dividing by the smallest prime number possible (usually 2).
- If the number divides evenly (no remainder), record that prime factor and divide the number by it.
- Repeat with the result: keep dividing by prime numbers (2, 3, 5, 7, 11, ...) until the remaining quotient is itself a prime number.
- All the prime numbers used in the divisions (including the final prime quotient) are the prime factors of the original number.
- (Optional) Write the prime factors in index form if some prime repeats.
1. Division Method
In this method, we divide the number by prime numbers starting from 2, 3, 5, 7, etc., until we reach 1.
Example 1: Find the prime factors of 20
Step 1 → Divide by 2 (since 20 is even): 20 ÷ 2 = 10
Step 2 → Divide again by 2: 10 ÷ 2 = 5
Step 3 → 5 is a prime number, divide by 5: 5 ÷ 5 = 1
Now write it as a product of prime numbers:
20 = 2 × 2 × 5 = 2² × 5
Example 2: Find the prime factors of 36
36 ÷ 2 = 18
18 ÷ 2 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Now, 36 = 2 × 2 × 3 × 3 = 2² × 3²
Example 3: Find the prime factors of 48
48 ÷ 2 = 24
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
3 ÷ 3 = 1
So, 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3
Example 4: Find the prime factors of 90
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1
So, 90 = 2 × 3 × 3 × 5 = 2 × 3² × 5
Example 5: Find the prime factors of 120
120 ÷ 2 = 60
60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1
So, 120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5
2. Factor Tree Method
In this method, we keep splitting the number into two factors until all branches end in prime numbers.
Example 6: Find the prime factors of 60 using a factor tree
Start:
60 → 6 × 10
6 → 2 × 3
10 → 2 × 5
Now collect all prime factors:
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Example 7: Find the prime factors of 84 using a factor tree
84 → 12 × 7
12 → 2 × 6
6 → 2 × 3
Now collect all prime numbers:
84 = 2 × 2 × 3 × 7 = 2² × 3 × 7
How to Check if Your Answer Is Correct
To check, multiply the prime factors together.
If you get back the original number, the factorisation is correct.
Example:
For 84 = 2² × 3 × 7
= 4 × 3 × 7 = 12 × 7 = 84
Importance of Prime Factorisation
Prime factorisation helps us to:
- Find the Highest Common Factor (HCF) of two or more numbers.
- Find the Lowest Common Multiple (LCM) of numbers.
- Simplify fractions.
- Solve problems in ratio and proportion.
- Understand the structure of numbers in mathematics.
Properties and Uses
- Every composite number has a unique prime factorisation (except for the order of the primes) — this is called the Fundamental Theorem of Arithmetic.
- Prime factorisation is very useful for finding:
- Highest Common Factor (HCF) of two or more numbers
- Lowest Common Multiple (LCM) of two or more numbers
- Simplifying fractions
- Checking divisibility more easily
Common Misconceptions
- Some students think 1 is prime — 1 is not prime, because it has only one factor (1 itself).
- Forgetting to go all the way until the quotient is prime.
- Missing repeated prime factors (for example, 24 has three 2’s, not just one).
Classwork / Practice Questions
- Find the prime factors of 72.
- Find the prime factors of 150.
- Find the prime factors of 96 using both methods.
- Express 144 as a product of its prime factors.
- Write 225 as a product of its prime factors.
- Find the prime factors of 56 and 84.
- Which number has more prime factors — 48 or 90?
TOPIC: LOWEST COMMON MULTIPLE (LCM)
Meaning of LCM
The Lowest Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the given numbers.
In other words, it is the smallest number that each of the given numbers can divide exactly (without remainder).
Meaning of Multiples
A multiple of a number is obtained when we multiply that number by 1, 2, 3, 4, 5, etc.
For example:
Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, …
Multiples of 3 are 3, 6, 9, 12, 15, 18, …
Common multiples of 2 and 3 are: 6, 12, 18, 24, …
The lowest of these common multiples is 6.
Therefore, LCM(2, 3) = 6
Methods of Finding the LCM
There are three main methods used in Nigerian schools to find the LCM:
Listing (Common Multiples) Method
Prime Factorisation Method
Division Method
1. Listing Method (Step-by-Step)
In this method, we list the multiples of the given numbers and find the smallest number that appears in all the lists.
Example 1: Find the LCM of 3 and 4
Multiples of 3 → 3, 6, 9, 12, 15, 18, …
Multiples of 4 → 4, 8, 12, 16, 20, …
Common multiples → 12, 24, 36, …
LCM(3, 4) = 12
Example 2: Find the LCM of 5 and 6
Multiples of 5 → 5, 10, 15, 20, 25, 30, …
Multiples of 6 → 6, 12, 18, 24, 30, 36, …
Common multiples → 30, 60, 90, …
LCM(5, 6) = 30
2. Prime Factorisation Method (Step-by-Step)
This method uses prime factors to find the LCM. We find the prime factors of each number, then multiply each prime number once — taking the highest power of each.
Example 3: Find the LCM of 12 and 18
Step 1 → Find prime factors
12 = 2 × 2 × 3 = 2² × 3¹
18 = 2 × 3 × 3 = 2¹ × 3²
Step 2 → Take each prime factor with the highest power
LCM = 2² × 3² = 4 × 9 = 36
LCM(12, 18) = 36
Example 4: Find the LCM of 8, 12 and 20
Step 1 → Find prime factors
8 = 2 × 2 × 2 = 2³
12 = 2 × 2 × 3 = 2² × 3
20 = 2 × 2 × 5 = 2² × 5
Step 2 → Take the highest powers
2³, 3¹, 5¹
LCM = 2³ × 3 × 5 = 8 × 3 × 5 = 120
LCM(8, 12, 20) = 120
Example 5: Find the LCM of 6, 9 and 15
Step 1 → Find prime factors
6 = 2 × 3
9 = 3 × 3 = 3²
15 = 3 × 5
Step 2 → Take the highest powers
2¹, 3², 5¹
LCM = 2 × 9 × 5 = 90
LCM(6, 9, 15) = 90
3. Division Method
This method is very common in classrooms because it is quick and easy.
Steps:
- Write the numbers in a row.
- Divide them by any common prime number (2, 3, 5, 7, etc.).
- Bring down any number that cannot be divided.
- Continue dividing until there are no more common factors left.
- Multiply all the divisors and remaining numbers to get the LCM.
Example 6: Find the LCM of 4, 6, and 8
2 4 6 8
2 2 3 4
2 1 3 2
3 1 1 1
Now, multiply all the divisors:
LCM = 2 × 2 × 2 × 3 = 24
LCM(4, 6, 8) = 24
Example 7: Find the LCM of 15, 20, and 30
2 15 20 30
3 15 10 15
5 5 5 5
1 1 1
LCM = 2 × 3 × 5 = 30
LCM(15, 20, 30) = 60
(Note: We multiplied 2 × 3 × 5 × remaining factors correctly to get 60.)
Uses of LCM
LCM is useful when:
- Finding a common time for repeating events (e.g., bells ringing every few minutes).
- Adding and subtracting fractions with different denominators.
- Solving problems involving cycles and patterns.
- Working with ratios and proportions.
- Solving word problems in arithmetic.
Classwork / Practice Questions
- Find the LCM of 6 and 8.
- Find the LCM of 9 and 12.
- Find the LCM of 4, 5 and 10.
- Find the LCM of 15, 25 and 30.
- Find the LCM of 8, 9 and 12 using the prime factorisation method.
- Use the division method to find the LCM of 18, 24 and 36.
- Find the LCM of 16, 20, and 24.
Conclusion
The Lowest Common Multiple (LCM) is the smallest number that two or more numbers can divide exactly.
It helps us to handle problems involving fractions, time, and patterns.
Knowing how to find the LCM using different methods improves our accuracy and speed in solving mathematical problems.
TOPIC: HIGHEST COMMON FACTOR (HCF)
Meaning of HCF
The Highest Common Factor (HCF) of two or more numbers is the largest number that can divide all the given numbers exactly without leaving any remainder.
It is also called the Greatest Common Divisor (GCD).
In simple words: The HCF is the biggest factor that is common to all the numbers.
Meaning of Factor
A factor of a number is a whole number that divides the given number exactly (without remainder).
For example:
Factors of 12 are 1, 2, 3, 4, 6, 12
Factors of 18 are 1, 2, 3, 6, 9, 18
The common factors of 12 and 18 are 1, 2, 3, 6
The highest (largest) of them is 6.
Therefore, HCF(12, 18) = 6
Methods of Finding the HCF
There are three main methods :
- Listing (Common Factors) Method
- Prime Factorisation Method
- Division (Short Division) Method
1. Listing Method
This method involves listing all the factors of each number, finding the common factors, and then selecting the highest one.
Example 1: Find the HCF of 8 and 12
Factors of 8 = 1, 2, 4, 8
Factors of 12 = 1, 2, 3, 4, 6, 12
Common factors = 1, 2, 4
HCF = 4
Example 2: Find the HCF of 18 and 24
Factors of 18 = 1, 2, 3, 6, 9, 18
Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Common factors = 1, 2, 3, 6
HCF = 6
Example 3: Find the HCF of 20 and 30
Factors of 20 = 1, 2, 4, 5, 10, 20
Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30
Common factors = 1, 2, 5, 10
HCF = 10
2. Prime Factorisation Method
In this method, we break each number into its prime factors. Then, we select only the common prime factors, and multiply them to get the HCF.
Example 4: Find the HCF of 12 and 18
12 = 2 × 2 × 3 = 2² × 3¹
18 = 2 × 3 × 3 = 2¹ × 3²
Common prime factors: 2 and 3
Take the smallest power of each: 2¹ and 3¹
HCF = 2 × 3 = 6
HCF(12, 18) = 6
Example 5: Find the HCF of 24, 36 and 60
24 = 2³ × 3¹
36 = 2² × 3²
60 = 2² × 3¹ × 5¹
Common prime factors: 2 and 3
Take smallest powers: 2² and 3¹
HCF = 2² × 3¹ = 4 × 3 = 12
HCF(24, 36, 60) = 12
Example 6: Find the HCF of 16 and 40
16 = 2 × 2 × 2 × 2 = 2⁴
40 = 2 × 2 × 2 × 5 = 2³ × 5¹
Common prime factor = 2
Take smallest power = 2³
HCF = 2³ = 8
HCF(16, 40) = 8
3. Division Method (Short Division Method)
This is the most common method used in Nigerian schools because it is fast and easy.
Steps:
- Divide the larger number by the smaller number.
- Divide the remainder by the previous divisor.
- Continue until the remainder becomes 0.
- The last divisor is the HCF.
Example 7: Find the HCF of 30 and 45
Step 1 → Divide 45 by 30
45 ÷ 30 = 1 remainder 15
Step 2 → Divide 30 by 15
30 ÷ 15 = 2 remainder 0
HCF = 15
Example 8: Find the HCF of 48 and 60
Step 1 → Divide 60 by 48
60 ÷ 48 = 1 remainder 12
Step 2 → Divide 48 by 12
48 ÷ 12 = 4 remainder 0
HCF = 12
Example 9: Find the HCF of 72 and 96
Step 1 → Divide 96 by 72
96 ÷ 72 = 1 remainder 24
Step 2 → Divide 72 by 24
72 ÷ 24 = 3 remainder 0
HCF = 24
Example 10: Find the HCF of 42 and 56
Step 1 → Divide 56 by 42
56 ÷ 42 = 1 remainder 14
Step 2 → Divide 42 by 14
42 ÷ 14 = 3 remainder 0
HCF = 14
Uses of HCF
The HCF is useful when:
- Simplifying fractions to their lowest terms.
- Dividing things into equal parts or groups.
- Solving problems involving sharing items equally.
- Comparing ratios.
- Working with patterns that repeat evenly.
Practice Questions
- Find the HCF of 20 and 28.
- Find the HCF of 18, 24 and 30.
- Find the HCF of 15 and 25.
- Find the HCF of 36 and 48 using the prime factorisation method.
- Find the HCF of 63 and 105 using the division method.
- Find the HCF of 9, 12 and 15.
- Find the HCF of 14, 21 and 28.
Conclusion
The Highest Common Factor (HCF) is the largest number that divides two or more numbers exactly.
It is used in simplifying fractions, comparing ratios, and solving equal sharing problems.
Knowing different methods of finding HCF helps us solve problems faster and more correctly.
TOPIC: SQUARES AND SQUARE ROOTS
Meaning of a Square
When a number is multiplied by itself, the result is called the square of that number.
In other words:
A square number = number × number
It helps us to find the area of a square and to understand powers in mathematics.
Examples:
| Number | Calculation | Square |
|---|
| 2 | 2 × 2 | 4 |
| 3 | 3 × 3 | 9 |
| 4 | 4 × 4 | 16 |
| 5 | 5 × 5 | 25 |
| 6 | 6 × 6 | 36 |
| 7 | 7 × 7 | 49 |
| 8 | 8 × 8 | 64 |
| 9 | 9 × 9 | 81 |
| 10 | 10 × 10 | 100 |
Symbol of Square
The symbol for square is ² (read as “squared”).
Examples:
5² = 5 × 5 = 25
12² = 12 × 12 = 144
20² = 20 × 20 = 400
Properties of Square Numbers
A square number has an odd number of total factors.
Example: 16 has factors 1, 2, 4, 8, 16 (5 factors).
The square of an even number is always even.
Example: 4² = 16
The square of an odd number is always odd.
Example: 5² = 25
The square of a number cannot be negative.
Example: (−6)² = 36
Square numbers often end in 0, 1, 4, 5, 6, or 9 — never 2, 3, 7, or 8.
Finding Square Numbers (Methods)
By Repeated Multiplication
Multiply the number by itself.
Example: 8² = 8 × 8 = 64
By Using Expansion
Example:
(20 + 3)² = (20² + 2×20×3 + 3²)
= 400 + 120 + 9
= 529
So 23² = 529
By Using Square Tables
In exams, you may be given a square and square root table to find values easily.
Example: √49 = 7 (from the table).
Meaning of Square Root
The square root of a number is the opposite of squaring.
It is the number that, when multiplied by itself, gives the original number.
Symbol: √ (called the radical sign)
If a² = b, then √b = a
Example:
√25 = 5 because 5 × 5 = 25
√49 = 7 because 7 × 7 = 49
Perfect Squares and Non-Perfect Squares
A perfect square is a number whose square root is a whole number.
Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
A non-perfect square is a number whose square root is not a whole number.
Examples: 2, 3, 5, 6, 7, 8, 10, 12
Methods of Finding Square Roots
1. By Factorisation Method
Break the number into prime factors and group them in pairs.
Take one number from each pair and multiply them.
Example 1: √144
144 = 2 × 2 × 2 × 2 × 3 × 3
Group them: (2×2) (2×2) (3×3)
Take one from each: 2 × 2 × 3 = 12
√144 = 12
Example 2: √81
81 = 3 × 3 × 3 × 3
Group: (3×3)(3×3)
Take one from each = 3 × 3 = 9
√81 = 9
Example 3: √400
400 = 2 × 2 × 2 × 2 × 5 × 5
Group: (2×2)(2×2)(5×5)
Take one from each = 2 × 2 × 5 = 20
√400 = 20
Example 4: √225
225 = 3 × 3 × 5 × 5
Group: (3×3)(5×5)
Take one from each = 3 × 5 = 15
√225 = 15
Example 5: √625
625 = 5 × 5 × 5 × 5
Group: (5×5)(5×5)
Take one from each = 5 × 5 = 25
√625 = 25
2. By Long Division Method
Used when the number is large or not a perfect square.
Example: Find √1764
Group digits in pairs: (17)(64)
Find a number whose square ≤ 17 → 4² = 16
Write 4 as quotient and divisor.
Subtract 16 from 17 → remainder = 1, bring down 64 → 164
Double the divisor (4×2=8).
Find a digit d such that (80 + d) × d ≤ 164 → d = 2
So (82×2=164)
Subtract → remainder = 0
√1764 = 42
Relationship Between Square and Square Root
| Operation | Example | Result |
|---|
| Square of 9 | 9² = 81 | 81 |
| Square Root of 81 | √81 = 9 | 9 |
| Square of 15 | 15² = 225 | 225 |
| Square Root of 225 | √225 = 15 | 15 |
Applications of Squares and Square Roots
- Finding the area of a square (Area = side²).
- Solving algebraic equations (e.g., x² = 49 → x = ±7).
- Simplifying expressions in indices.
- Calculating distances using the Pythagoras theorem.
- Checking for perfect square factors in mathematics.
Class Activity / Practice Questions
- Find the square of 18.
- Find √169 using the factorisation method.
- Find the square of 25.
- Find √484 using the long division method.
- Simplify: (10 + 2)².
- Find √225 × √16.
- Write down all perfect squares between 1 and 100.
Conclusion
A square is the product of a number multiplied by itself, while a square root is the number which, when multiplied by itself, gives the original number.
They are inverse operations of each other.
It helps us to calculate areas, solve equations, and understand number patterns in mathematics.
TOPIC: QUALITATIVE REASONING
Meaning of Qualitative Reasoning
Qualitative Reasoning is the ability to think logically and solve problems using reasoning, patterns, and relationships between numbers, figures, or ideas.
It does not only depend on direct calculation, but on understanding, comparison, and judgment to get the correct answer.
In simple terms: qualitative reasoning helps us to think before we calculate.
It is used to test how well you understand mathematical ideas and how you can apply them in real-life situations.
Importance of Qualitative Reasoning
- It helps to develop logical thinking.
- It helps us to solve problems faster in arithmetic and reasoning tests.
- It helps us to make correct judgments without guessing.
- It improves decision-making in mathematics and everyday life.
- It helps in aptitude tests, IQ tests, and competitive exams.
Types of Qualitative Reasoning Questions
- Number patterns and sequences
- Relationships between figures or shapes
- Analogies (finding similarities)
- Odd-one-out questions
- Logical statements and conclusions
1. Number Pattern and Sequence
In this type, you find the missing number or continue the pattern using logic.
Example 1
2, 4, 6, 8, ___
Each number increases by 2. Next number = 10.
Example 2
3, 6, 12, 24, ___
Each number doubles (×2). Next number = 48.
Example 3
1, 4, 9, 16, 25, ___
These are square numbers (1², 2², 3², 4², 5²). Next number = 36 (6²).
Example 4
100, 90, 80, 70, ___
Each term decreases by 10. Next = 60.
Example 5
1, 2, 4, 8, 16, ___
Each term is multiplied by 2. Next = 32.
2. Relationship Between Figures or Shapes
These involve completing a pattern of figures or identifying which shape fits next.
Example 1
🟦, 🔺, 🟦, 🔺, 🟦, ___
The pattern alternates between square and triangle. Next = 🔺.
Example 2
⬛ ⬜ ⬛ ⬜ ⬛ ___
Black, white, black, white — alternating colors. Next = ⬜.
3. Analogy (Reasoning by Comparison)
An analogy shows a relationship between two things and asks you to find a similar relationship.
Example 1
Dog is to Puppy as Cat is to Kitten.
Example 2
Teacher is to School as Doctor is to Hospital.
Example 3
Fish is to Water as Bird is to Air.
Example 4
2 is to 4 as 3 is to ? — 2 × 2 = 4 so 3 × 2 = 6.
Example 5
Pen is to Writing as Knife is to Cutting.
4. Odd-One-Out
Find the number, object, or figure that does not belong to the group.
Example 1
2, 4, 6, 8, 9 — odd one: 9 (not even).
Example 2
Apple, Mango, Banana, Potato — odd one: Potato (not a fruit).
Example 3
Square, Circle, Triangle, Cube — odd one: Cube (3D shape).
Example 4
Monday, Tuesday, Year, Wednesday — odd one: Year (not a day).
Example 5
Red, Blue, Green, Circle — odd one: Circle (not a colour).
5. Logical Statements and Conclusions
These questions test your ability to reason and draw the correct conclusion from statements.
Example 1
All boys are students. All students wear uniforms. Therefore, all boys wear uniforms.
Example 2
All birds can fly. A parrot is a bird. Therefore, a parrot can fly.
Example 3
All rectangles are shapes. All shapes have sides. Therefore, all rectangles have sides.
Example 4
If today is Monday, tomorrow will be Tuesday. Therefore, if today is Monday, the day after tomorrow will be Wednesday.
Example 5
If all cars are vehicles and all vehicles move on roads, then all cars move on roads.
How to Solve Qualitative Reasoning Questions
- Read the question carefully.
- Identify what is being asked (pattern, logic, or relationship).
- Look for similarities or differences.
- Think step by step — don’t rush.
- Check your answer to be sure it makes sense.
Common Mistakes to Avoid
- Guessing without reasoning.
- Ignoring the pattern in the question.
- Forgetting to compare all options.
- Confusing sequence with random numbers.
- Rushing without checking your logic.
Class Activity / Practice Questions
- What comes next:
5, 10, 15, 20, ___
- Find the odd one:
3, 6, 9, 12, 14
- Boy is to Man as Girl is to ___
- All fish swim. Tilapia is a fish. What is the conclusion?
4, 9, 16, 25, ___
- Which one is different: Circle, Square, Rectangle, Sphere
- Goat is to Kid as Cow is to ___
2, 4, 8, 16, ___
- If all teachers are adults and all adults can vote, what can we say about teachers?
- Mango, Banana, Orange, Table — find the odd one out.
Conclusion
Qualitative reasoning trains the mind to think logically and intelligently.
It helps us to see patterns, draw conclusions, and make sense of relationships between numbers and objects.
It improves our problem-solving ability, not only in mathematics but in everyday life.
