Learning Objectives
Chapter Overview
Key skills you will develop by the end of Chapter 1: Primes, HCF, LCM and Integers.
Identify prime and composite numbers using divisibility tests and trial division
Perform prime factorization using the factor tree and repeated division methods
Calculate HCF and LCM of two or more numbers using both listing and division methods
Solve real-world word problems involving HCF and LCM applications
Understand negative numbers and position integers accurately on a number line
Execute all four arithmetic operations on integers, fractions, and decimals fluently
Mathematics: Primes, HCF, LCM and Integers
Complete chapter notes with worked examples, practice problems and vocabulary — PDF format
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Part 1
Prime Numbers & Factorization
Understanding primes, composites, divisibility, and prime factorization.
Prime and Composite Numbers
1. What is a prime number? What is a composite number?
• Define prime and composite numbers with examples• How many factors does a prime number have?
A Prime Number is a whole number with exactly 2 different factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, …
A Composite Number is a whole number with more than 2 different factors. Examples: 4, 6, 8, 9, 10, 12, …
A Composite Number is a whole number with more than 2 different factors. Examples: 4, 6, 8, 9, 10, 12, …
📝 Note: 1 has only ONE factor → neither prime nor composite. 0 is also neither prime nor composite. The number 2 is the ONLY even prime number.
2. What is the Sieve of Eratosthenes? List all primes up to 100.
• How do you find all prime numbers up to 100?
The Sieve of Eratosthenes is a method to find all primes up to any limit. Circle 2 (first prime), cross out all multiples of 2. Circle the next uncrossed number, cross out all its multiples. Repeat.
Primes up to 100: 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
Primes up to 100: 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
3. What are the divisibility tests for 2, 3, 5, 7, and 11?
• How do you check if a large number is divisible by 3?• What is the quick test for divisibility by 5?
Divisible by 2: Last digit is 0, 2, 4, 6, or 8
Divisible by 3: Sum of all digits is divisible by 3 (e.g. 387: 3+8+7=18 ✓)
Divisible by 5: Last digit is 0 or 5
Divisible by 7: Double the last digit, subtract from the rest; check if result is divisible by 7
Divisible by 11: Subtract sum of even-position digits from odd-position digits; result must be 0 or multiple of 11
Divisible by 3: Sum of all digits is divisible by 3 (e.g. 387: 3+8+7=18 ✓)
Divisible by 5: Last digit is 0 or 5
Divisible by 7: Double the last digit, subtract from the rest; check if result is divisible by 7
Divisible by 11: Subtract sum of even-position digits from odd-position digits; result must be 0 or multiple of 11
4. What is Trial Division? How do you test if a number is prime?
• Test whether 47 is prime or composite
To test if n is prime: calculate √n, then test divisibility by all primes up to √n.
Example — Test 47: √47 ≈ 6.86. Test primes {2, 3, 5}: 47 is odd ✗2; digit sum 11 ✗3; ends in 7 ✗5. None divide 47 exactly. ∴ 47 is PRIME.
Example — Test 47: √47 ≈ 6.86. Test primes {2, 3, 5}: 47 is odd ✗2; digit sum 11 ✗3; ends in 7 ✗5. None divide 47 exactly. ∴ 47 is PRIME.
Prime Factorization & Index Notation
5. What is prime factorization? Explain with an example.
• Find the prime factorization of 60• What is the factor tree method?
Every composite number can be written uniquely as a product of prime numbers. This is its prime factorization.
Example — Prime factorization of 60:
Method 1 (Factor Tree): 60 = 2×30 = 2×2×15 = 2×2×3×5
Method 2 (Repeated Division): 60÷2=30, 30÷2=15, 15÷3=5, 5÷5=1
Example — Prime factorization of 60:
Method 1 (Factor Tree): 60 = 2×30 = 2×2×15 = 2×2×3×5
Method 2 (Repeated Division): 60÷2=30, 30÷2=15, 15÷3=5, 5÷5=1
∴ 60 = 2² × 3 × 5
6. What is index notation? How is it used with prime factorization?
• What does 5³ mean?
When the same number is multiplied repeatedly, we use index notation:
5 × 5 = 5² (read: "5 squared")
5 × 5 × 5 = 5³ (read: "5 cubed")
5 × 5 × 5 × 5 = 5⁴ (read: "5 to the power 4")
The small raised number is the index (exponent) and the large number is the base.
5 × 5 = 5² (read: "5 squared")
5 × 5 × 5 = 5³ (read: "5 cubed")
5 × 5 × 5 × 5 = 5⁴ (read: "5 to the power 4")
The small raised number is the index (exponent) and the large number is the base.
Part 2
HCF & LCM — Both Methods
Finding HCF and LCM using listing and division methods, with real-world applications.
Highest Common Factor (HCF)
1. What is the HCF? How do you find it using the listing method?
• Define HCF• Find the HCF of 12 and 18 by listing
The Highest Common Factor (HCF) of two or more numbers is the largest factor shared by all of them.
Listing Method: List all factors of each number, then find the largest one they share.
Factors of 12: {1, 2, 3, 4, 6, 12}
Factors of 18: {1, 2, 3, 6, 9, 18}
Common factors: {1, 2, 3, 6} → HCF = 6
Listing Method: List all factors of each number, then find the largest one they share.
Factors of 12: {1, 2, 3, 4, 6, 12}
Factors of 18: {1, 2, 3, 6, 9, 18}
Common factors: {1, 2, 3, 6} → HCF = 6
2. How do you find HCF using prime factorization (division method)?
• Find HCF of 36 and 48 using prime factorization
Prime Factorization Method:
36 = 2² × 3² and 48 = 2⁴ × 3
Take the lowest power of each common prime factor:
Common primes: 2 and 3 → 2² × 3¹
36 = 2² × 3² and 48 = 2⁴ × 3
Take the lowest power of each common prime factor:
Common primes: 2 and 3 → 2² × 3¹
HCF(36, 48) = 4 × 3 = 12
Lowest Common Multiple (LCM)
3. What is the LCM? How do you find it using the listing method?
• Define LCM• Find LCM of 4 and 6 by listing multiples
The Lowest Common Multiple (LCM) of two or more numbers is the smallest multiple shared by all of them.
Multiples of 4: 4, 8, 12, 16, 20 …
Multiples of 6: 6, 12, 18, 24 …
Smallest common multiple → LCM = 12
Multiples of 4: 4, 8, 12, 16, 20 …
Multiples of 6: 6, 12, 18, 24 …
Smallest common multiple → LCM = 12
4. How do you find LCM using prime factorization?
• Find LCM of 12 and 18 using prime factors
12 = 2² × 3 and 18 = 2 × 3²
Take the highest power of each prime factor that appears:
Primes used: 2 and 3 → 2² × 3²
Take the highest power of each prime factor that appears:
Primes used: 2 and 3 → 2² × 3²
LCM(12, 18) = 4 × 9 = 36
5. How do you decide whether to use HCF or LCM in a word problem?
• When do we use HCF in real life?• When is LCM used in word problems?
Use HCF when: you need to DIVIDE or SPLIT things into the largest equal groups (e.g., cutting a ribbon into equal pieces, sharing items equally).
Use LCM when: you need to find when two events MEET AGAIN or a minimum amount that satisfies multiple conditions (e.g., buses arriving at the same time, buying exact packs).
Use LCM when: you need to find when two events MEET AGAIN or a minimum amount that satisfies multiple conditions (e.g., buses arriving at the same time, buying exact packs).
Part 3
Integers & Operations
Negative numbers, number lines, and arithmetic with integers.
Integers and the Number Line
1. What are integers? What are natural, whole, and rational numbers?
• Define natural numbers, whole numbers, integers• Is 0 a positive or negative integer?
Natural Numbers (N): {1, 2, 3, 4, …} — counting numbers from 1
Whole Numbers (W): {0, 1, 2, 3, …} — natural numbers plus zero
Integers (Z): {…, −3, −2, −1, 0, 1, 2, 3, …} — all whole numbers and their negatives
Rational Numbers (Q): Any number expressible as a/b where b ≠ 0
Whole Numbers (W): {0, 1, 2, 3, …} — natural numbers plus zero
Integers (Z): {…, −3, −2, −1, 0, 1, 2, 3, …} — all whole numbers and their negatives
Rational Numbers (Q): Any number expressible as a/b where b ≠ 0
📝 N ⊂ W ⊂ Z ⊂ Q. Zero is an integer but is neither positive nor negative.
2. How do you add and subtract integers?
• What is the rule for adding two negative numbers?• Calculate: (−3) + (−5); (−4) − (−2)
Adding integers: Same signs → add and keep the sign. Different signs → subtract smaller from larger and keep the sign of the larger number.
(−3) + (−5) = −8 (both negative: add, keep −)
(+6) + (−4) = +2 (different: 6−4=2, keep +)
Subtracting integers: Change subtraction to addition, then flip the sign of the second number.
(−4) − (−2) = (−4) + (+2) = −2
(−3) + (−5) = −8 (both negative: add, keep −)
(+6) + (−4) = +2 (different: 6−4=2, keep +)
Subtracting integers: Change subtraction to addition, then flip the sign of the second number.
(−4) − (−2) = (−4) + (+2) = −2
3. What are the rules for multiplying and dividing integers?
• What is positive × negative?• What is negative ÷ negative?
Same signs → Positive result:
(+) × (+) = + and (−) × (−) = +
Different signs → Negative result:
(+) × (−) = − and (−) × (+) = −
The same rules apply to division (÷).
(+) × (+) = + and (−) × (−) = +
Different signs → Negative result:
(+) × (−) = − and (−) × (+) = −
The same rules apply to division (÷).
Positive × Positive = Positive (+)
Negative × Negative = Positive (+)
Positive × Negative = Negative (−)
Negative × Negative = Positive (+)
Positive × Negative = Negative (−)
Vocabulary
Key Terms
Important vocabulary for Chapter 1.
| Term | Simple Meaning |
|---|---|
| Prime Number | A whole number with exactly 2 factors: 1 and itself (e.g. 2, 3, 5, 7) |
| Composite Number | A whole number with more than 2 factors (e.g. 4, 6, 8, 9) |
| Factor | A whole number that divides into another number exactly, with no remainder |
| Prime Factorization | Writing a number as a product of its prime factors only |
| Index Notation | Writing repeated multiplication using a base and exponent (e.g. 2³ = 2×2×2) |
| HCF | Highest Common Factor — the largest factor shared by two or more numbers |
| LCM | Lowest Common Multiple — the smallest multiple shared by two or more numbers |
| Integer | Any whole number including negative numbers and zero: {…, −2, −1, 0, 1, 2, …} |
| Rational Number | Any number that can be written as a fraction a/b where b ≠ 0 |
| Sieve of Eratosthenes | A method to find all prime numbers up to a given limit by eliminating multiples |
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