Factors and Multiples
1. Factors
A FACTOR of a number divides it EXACTLY — leaving NO remainder.
'Every number has 1 and itself as a factor. Every factor is SMALLER than or EQUAL to the number.'
Finding Factors
To find all factors of 36:
36 ÷ 1 = 36 (1, 36) 36 ÷ 2 = 18 (2, 18) 36 ÷ 3 = 12 (3, 12) 36 ÷ 4 = 9 (4, 9) 36 ÷ 6 = 6 (6 — the square root)
Factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36}
Properties of Factors
| Property | Example |
|---|---|
| 1 is a factor of EVERY number | 1 × 24 = 24 |
| Every number is a factor of ITSELF | 24 × 1 = 24 |
| A factor is always SMALLER than or EQUAL to the number | Factors of 18 are ≤ 18 |
| The number of factors is FINITE | 18 has 6 factors |
2. Multiples
A MULTIPLE is obtained by multiplying a number by a WHOLE number.
'Multiples are INFINITE — you can keep multiplying forever. There is NO largest multiple of any number except zero.'
Finding Multiples
Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, ...
7 × 1 = 7 7 × 2 = 14 7 × 3 = 21 ... and so on
Properties of Multiples
| Property | Example |
|---|---|
| Every number is a multiple of ITSELF | The first multiple of any number is the number itself |
| Multiples are INFINITE | There is no end to multiples |
| Zero is a multiple of EVERY number | 0 = 6 × 0 |
| The smallest POSITIVE multiple is the number itself | First multiple of 12 is 12 |
Difference Between Factors and Multiples
| Factor | Multiple |
|---|---|
| Divides the number exactly | Is divisible by the number |
| Finite in count | Infinite in count |
| Always ≤ the number | Always ≥ the number |
| Example: Factors of 12 are 1, 2, 3, 4, 6, 12 | Example: Multiples of 12 are 12, 24, 36, 48, ... |
3. Prime and Composite Numbers
Prime Numbers
A PRIME number has EXACTLY TWO factors: 1 and itself.
Prime numbers up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
'The number 2 is the ONLY even prime number. All other even numbers have 2 as a factor, so they are composite.'
Composite Numbers
A COMPOSITE number has MORE than two factors.
Composite numbers up to 20: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20.
Special Numbers
| Number | Type | Reason |
|---|---|---|
| 0 | Neither prime nor composite | 0 has INFINITE factors |
| 1 | Neither prime nor composite | 1 has ONLY ONE factor (1) |
| 2 | Prime | Only factors: 1 and 2 |
| 3 | Prime | Only factors: 1 and 3 |
4. Divisibility Tests
| Divisible By | Rule | Example |
|---|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) | 734 ends in 4 → YES |
| 3 | Sum of digits is divisible by 3 | 471: 4 + 7 + 1 = 12, 12 ÷ 3 = 4 → YES |
| 4 | Last TWO digits divisible by 4 | 732: last two digits 32, 32 ÷ 4 = 8 → YES |
| 5 | Last digit is 0 or 5 | 675 ends in 5 → YES |
| 6 | Divisible by BOTH 2 and 3 | 342: ends in 2 (✓), 3+4+2=9 ÷ 3 (✓) → YES |
| 8 | Last THREE digits divisible by 8 | 7,128: 128 ÷ 8 = 16 → YES |
| 9 | Sum of digits is divisible by 9 | 4,257: 4+2+5+7=18, 18 ÷ 9 = 2 → YES |
| 10 | Last digit is 0 | 3,890 ends in 0 → YES |
'A number divisible by 2 is EVEN. A number NOT divisible by 2 is ODD. You can combine rules — if a number passes BOTH the 2 and 3 rules, it is divisible by 6.'
5. HCF — Highest Common Factor
The HCF of two or more numbers is the LARGEST factor that divides ALL the numbers.
Method 1: Prime Factorisation
Find HCF of 24 and 36:
24 = 2 × 2 × 2 × 3 36 = 2 × 2 × 3 × 3
Common factors: 2 × 2 × 3 = 12
HCF(24, 36) = 12
Method 2: Long Division Method
'Divide the larger number by the smaller. Then divide the divisor by the remainder. Repeat until remainder is 0. The LAST divisor is the HCF.'
| Step | Calculation | Remainder |
|---|---|---|
| 1 | 36 ÷ 24 | 12 |
| 2 | 24 ÷ 12 | 0 |
HCF = 12
6. LCM — Lowest Common Multiple
The LCM of two or more numbers is the SMALLEST number that is a multiple of ALL the numbers.
Method: Prime Factorisation
Find LCM of 6 and 8:
6 = 2 × 3 8 = 2 × 2 × 2
Take the HIGHEST power of each prime factor: 2³ × 3 = 8 × 3 = 24
LCM(6, 8) = 24
Relation Between HCF and LCM
'For any two numbers: LCM × HCF = Product of the numbers.'
Check: LCM(6, 8) × HCF(6, 8) = 24 × 2 = 48 = 6 × 8 ✓
7. Co-Prime Numbers
Two numbers are CO-PRIME if their HCF is 1.
| Example | HCF | Co-prime? |
|---|---|---|
| 8 and 15 | 1 | YES |
| 6 and 10 | 2 | NO |
| 7 and 11 | 1 | YES |
| 14 and 21 | 7 | NO |
'Co-prime numbers do NOT have to be prime themselves. 8 and 15 are both composite — but they are co-prime because they share NO common factor except 1.'
Key Facts to Remember
- Every number greater than 1 is either PRIME or COMPOSITE.
- 1 is neither prime nor composite.
- All even numbers are divisible by 2.
- The HCF of co-prime numbers is ALWAYS 1.
- The LCM of two numbers is ALWAYS greater than or equal to each number.
- 'Knowing divisibility rules saves TIME in exams — you will not need to do long division for every test.'
Common Mistakes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| Saying 1 is prime | 1 has only ONE factor, not two | 1 is neither prime nor composite |
| Forgetting 2 is prime | 2 is even, so some think it is composite | 2 has exactly two factors: 1 and 2 |
| Confusing HCF and LCM | HCF is the GREATEST common factor; LCM is the LEAST common multiple | HCF ≤ both numbers; LCM ≥ both numbers |
| Divisibility by 6 — only checking last digit | A number must pass BOTH 2 and 3 tests | 34 is even but 3+4=7 (not divisible by 3), so NOT divisible by 6 |
Exam Focus (ICSE Class 5)
| Topic | Marks (Typical) | Question Type |
|---|---|---|
| Prime and composite identification | 2-3 marks | List primes/composites in a range |
| Divisibility tests | 3-4 marks | State whether divisible and why |
| HCF by prime factorisation | 3-4 marks | Find HCF of given numbers |
| LCM by prime factorisation | 3-4 marks | Find LCM of given numbers |
| Word problems (HCF/LCM application) | 4-5 marks | Real-life scenarios |
Self-Test: 5 Questions
Q1. List all prime numbers between 50 and 70.
Q2. Check if 7,245 is divisible by 3, 6, and 9. Show your working.
Q3. Find the HCF of 48 and 72 using prime factorisation.
Q4. Find the LCM of 12, 15, and 20.
Q5. Three bells ring at intervals of 4, 6, and 9 minutes. After how many minutes will they ring together again?
Answers
A1. 53, 59, 61, 67.
A2. Sum of digits = 7 + 2 + 4 + 5 = 18. 18 ÷ 3 = 6 → divisible by 3. 18 ÷ 9 = 2 → divisible by 9. It ends in 5 (not even), so NOT divisible by 2 → NOT divisible by 6.
A3. 48 = 2 × 2 × 2 × 2 × 3. 72 = 2 × 2 × 2 × 3 × 3. Common: 2 × 2 × 2 × 3 = 24. HCF = 24.
A4. 12 = 2² × 3. 15 = 3 × 5. 20 = 2² × 5. LCM = 2² × 3 × 5 = 60.
A5. Find LCM of 4, 6, and 9. 4 = 2², 6 = 2 × 3, 9 = 3². LCM = 2² × 3² = 36. They ring together every 36 minutes.
