Factors
Find factors by pairing; common factors of two numbers.
Factors
Factors of a Number
What you'll learn
- A factor divides a number exactly (remainder 0).
- Find all factors by multiplication pairs.
- Find common factors of two numbers for sharing problems.
Key concepts
Level 1 — Definition
If 12 ÷ 3 = 4 with no remainder, then 3 and 4 are factors of 12.
Verbal: Factors are exact divisors — they split a number into equal whole parts.
Level 2 — Pair method for 24
| × | Pair | Factors |
|---|---|---|
| 1 | 1 × 24 | 1, 24 |
| 2 | 2 × 12 | 2, 12 |
| 3 | 3 × 8 | 3, 8 |
| 4 | 4 × 6 | 4, 6 |
| 6 | 6 × 6 | stop — repeat |
All factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Level 3 — Common factors
| Number | Factors |
|---|---|
| 12 | 1, 2, 3, 4, 6, 12 |
| 18 | 1, 2, 3, 6, 9, 18 |
Common factors of 12 and 18: 1, 2, 3, 6. HCF = 6.
Level 4 — Indian context
48 Diwali diyas arranged in equal rows on a rangoli — possible rows: 2, 3, 4, 6, 8, 12, 16, 24 (all factors of 48).
NCERT anchor: Math-Magic 4, Ch 11 — Tables and Shares; Ch 14 — Play with Patterns (equal groups and arrays)
Worked example
Find all factors of 36
Step 1 — Pairs: 1×36, 2×18, 3×12, 4×9, 6×6.
Step 2 — List: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Step 3 — Verify: 36 ÷ 9 = 4 ✓
Answer: 9 factors
Common factors of 20 and 30
Step 1 — Factors of 20: 1, 2, 4, 5, 10, 20.
Step 2 — Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
Step 3 — Common: 1, 2, 5, 10.
Answer: HCF = 10
Common mistakes
| Mistake | Why it happens | Fix |
|---|---|---|
| 5 is a factor of 12 | Any small number divides | 12 ÷ 5 = 2 remainder 2 — not a factor |
| Forgetting 1 and the number itself | Only listing 'middle' factors | Every whole number has factors 1 and itself |
| Confusing factor with multiple | Mixing × and ÷ language | 3 is a factor of 12; 12 is a multiple of 3 |
| Missing pairs after 6×6 | Stopping too early | Continue until pairs repeat |
Quick check
- List all factors of 20.
- Is 7 a factor of 42? (yes — 42 ÷ 7 = 6)
- Common factor of 15 and 25? (5)
- Stretch: A farmer has 60 mango saplings. He wants equal rows with more than 1 row and fewer than 60 rows. How many different row counts are possible? (8: 2,3,4,5,6,10,12,15,20,30 — actually count factors excluding 1 and 60 = 8 ways)
Revision tip: Write multiplication pairs in two columns that meet in the middle — nothing gets missed.
Open the Practice tab for graded questions on Factors of a Number.
Key Takeaways (TL;DR)
- What you'll learn
- Key concepts
- Worked example
- Common mistakes
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