Factors and Multiples
1. Factors
A FACTOR of a number divides it EXACTLY — with NO remainder.
'2 is a factor of 6 because 6 ÷ 2 = 3 — no remainder. 4 is NOT a factor of 6 because 6 ÷ 4 = 1 remainder 2.'
How to Find Factors:
List ALL pairs of numbers that multiply to give the number.
Example — Factors of 12:
| Pair | Calculation |
|---|---|
| 1 × 12 | 1 and 12 |
| 2 × 6 | 2 and 6 |
| 3 × 4 | 3 and 4 |
Factors of 12: 1, 2, 3, 4, 6, 12
Example — Factors of 24:
1 × 24, 2 × 12, 3 × 8, 4 × 6
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Key Facts about Factors:
- 'EVERY number has 1 and ITSELF as factors.'
- 'The smallest factor of ANY number is 1.'
- 'The largest factor of a number is the number itself.'
- 'A number that has ONLY TWO factors (1 and itself) is called a PRIME number.'
2. Multiples
A MULTIPLE is the product of a number and any WHOLE number.
'Multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30... (3 × 1, 3 × 2, 3 × 3, etc.)'
How to Find Multiples:
Multiply the number by 1, 2, 3, 4, 5... and so on.
Example — Multiples of 5:
| Number | Calculation | Multiple |
|---|---|---|
| 5 × 1 | 5 | 5 |
| 5 × 2 | 10 | 10 |
| 5 × 3 | 15 | 15 |
| 5 × 4 | 20 | 20 |
| 5 × 5 | 25 | 25 |
Key Facts about Multiples:
- 'Multiples of a number are UNLIMITED — they go on forever.'
- 'The smallest multiple of any number is the number itself.'
- 'Every number is a multiple of 1.'
- '0 is a multiple of EVERY number (0 × any number = 0).'
3. Factors vs Multiples — The Difference
| Factors | Multiples |
|---|---|
| FINITE (limited) | INFINITE (unlimited) |
| Smaller than or equal to the number | Larger than or equal to the number |
| Divide the number exactly | Are divided by the number exactly |
| Example: Factors of 12 → 1, 2, 3, 4, 6, 12 | Example: Multiples of 12 → 12, 24, 36, 48, 60... |
'Think of it this way: FACTORS are the numbers that DIVIDE a number. MULTIPLES are the numbers you GET when you MULTIPLY a number.'
4. Prime and Composite Numbers
Prime Numbers:
A PRIME number has EXACTLY TWO factors: 1 and itself.
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
| Number | Factors | Prime? |
|---|---|---|
| 2 | 1, 2 | YES |
| 3 | 1, 3 | YES |
| 5 | 1, 5 | YES |
| 7 | 1, 7 | YES |
| 11 | 1, 11 | YES |
Composite Numbers:
A COMPOSITE number has MORE than two factors.
Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20...
| Number | Factors | Composite? |
|---|---|---|
| 4 | 1, 2, 4 | YES |
| 6 | 1, 2, 3, 6 | YES |
| 8 | 1, 2, 4, 8 | YES |
| 9 | 1, 3, 9 | YES |
| 10 | 1, 2, 5, 10 | YES |
Special Numbers:
- 1 is NEITHER prime NOR composite. It has only ONE factor (1).
- 2 is the SMALLEST and the ONLY EVEN prime number.
- 0 is neither prime nor composite.
'Remember: Prime numbers have EXACTLY 2 factors. Composite numbers have MORE than 2 factors. The number 1 is special — it is neither!'
5. Divisibility Tests
Divisible by 2:
A number is divisible by 2 if its LAST digit is EVEN (0, 2, 4, 6, or 8).
Examples: 34, 568, 7,890, 12,346 → YES Not divisible: 35, 789, 12,347
Divisible by 3:
A number is divisible by 3 if the SUM of its digits is divisible by 3.
Example: 123 → 1 + 2 + 3 = 6 → 6 is divisible by 3 → YES Example: 457 → 4 + 5 + 7 = 16 → 16 is NOT divisible by 3 → NO
Divisible by 5:
A number is divisible by 5 if its LAST digit is 0 or 5.
Examples: 25, 100, 345, 1,230 → YES Not divisible: 32, 128, 4,567
Divisible by 10:
A number is divisible by 10 if its LAST digit is 0.
Examples: 20, 100, 450, 1,230 → YES Not divisible: 25, 105, 455
| Test | Rule | Example |
|---|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) | 346 ✓ |
| 3 | Sum of digits ÷ 3 | 123 → 1+2+3=6 ✓ |
| 5 | Last digit is 0 or 5 | 345 ✓ |
| 10 | Last digit is 0 | 450 ✓ |
6. HCF — Highest Common Factor
The HCF of two numbers is the LARGEST number that divides BOTH numbers exactly.
Method — Listing Factors:
Find HCF of 12 and 18.
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
Example 2:
Find HCF of 15 and 25.
Factors of 15: 1, 3, 5, 15 Factors of 25: 1, 5, 25
Common factors: 1, 5 HCF = 5
'The HCF is the BIGGEST number that can divide BOTH numbers without leaving a remainder.'
7. LCM — Lowest Common Multiple
The LCM of two numbers is the SMALLEST number that is a MULTIPLE of BOTH numbers.
Method — Listing Multiples:
Find LCM of 4 and 6.
Multiples of 4: 4, 8, 12, 16, 20, 24... Multiples of 6: 6, 12, 18, 24, 30...
Common multiples: 12, 24, 36... LCM = 12
Example 2:
Find LCM of 3 and 5.
Multiples of 3: 3, 6, 9, 15, 18, 21, 24, 27, 30... Multiples of 5: 5, 10, 15, 20, 25, 30...
Common multiples: 15, 30... LCM = 15
'The LCM is the SMALLEST number that BOTH numbers divide into exactly.'
8. Common Mistakes
- Confusing factors and multiples: 'Factors are SMALLER than or EQUAL to the number. Multiples are LARGER than or EQUAL to the number.'
- Calling 1 a prime number: '1 is NOT prime. It has only ONE factor. Prime numbers must have EXACTLY TWO factors.'
- Not checking the sum for divisibility by 3: 'Keep adding digits until you get a single digit. If that digit is 3, 6, or 9, the number is divisible by 3.'
- HCF is always SMALLER or EQUAL to the numbers. LCM is always LARGER or EQUAL to the numbers.
9. Key Facts to Remember
- 'Every number has exactly ONE factor pair for each of its factors.'
- 'The only EVEN prime number is 2. All other even numbers are composite.'
- 'A number divisible by 2 and 3 is also divisible by 6.'
- 'HCF ≤ each of the numbers ≤ LCM.'
- 'All numbers are divisible by 1.'
10. Self-Test
Q1: List all factors of 36.
Q2: List the first 5 multiples of 7.
Q3: Is 51 a prime or composite number? Why?
Q4: Check divisibility: Is 4,560 divisible by (a) 2 (b) 3 (c) 5 (d) 10?
Q5: Find the HCF of 18 and 27.
Q6: Find the LCM of 6 and 8.
Q7: True or False: All prime numbers are odd.
Q8: Write the smallest 4-digit number that is divisible by 5.
Answers:
A1: 1, 2, 3, 4, 6, 9, 12, 18, 36 A2: 7, 14, 21, 28, 35 A3: Composite. Factors of 51: 1, 3, 17, 51 (more than 2 factors). A4: (a) Yes — last digit 0 (even) (b) Yes — 4+5+6+0=15, 15÷3=5 (c) Yes — last digit 0 (d) Yes — last digit 0 A5: 9 A6: 24 A7: False — 2 is a prime number and it is even. A8: 1,000
