Factors of a Number
Factors are numbers that divide a given number exactly (with remainder 0). Finding factors is a fundamental skill in Class 4 that connects multiplication and division.
Understanding factors helps in simplifying fractions, finding common factors, and solving many types of maths problems.
What is Factors of a Number - Class 4 Maths (Factors and Multiples)?
A factor of a number is a whole number that divides it exactly, leaving no remainder.
For example, the factors of 12 are: 1, 2, 3, 4, 6, 12 — because each of these divides 12 with remainder 0.
If a × b = n, then both a and b are factors of n.
Factors of a Number Formula
Method to find all factors of a number:
- Start with 1 and the number itself (always factors).
- Try dividing by 2, 3, 4, 5, ... in order.
- If the division is exact, both the divisor and the quotient are factors.
- Stop when the divisor exceeds the quotient (you have found all pairs).
Types and Properties
Important facts about factors:
- 1 is a factor of every number.
- Every number is a factor of itself.
- Factors always come in pairs. If 3 × 4 = 12, then 3 and 4 are both factors of 12.
- The number of factors is always finite (unlike multiples, which go on forever).
- A perfect square has an odd number of factors (e.g., 9 has factors 1, 3, 9).
Solved Examples
Example 1: Example 1: Factors of 12
Problem: Find all factors of 12.
Solution:
12 = 1 × 12 → factors: 1, 12
12 = 2 × 6 → factors: 2, 6
12 = 3 × 4 → factors: 3, 4
4 > 3, so we stop (next would repeat).
Answer: Factors of 12 = 1, 2, 3, 4, 6, 12 (6 factors)
Example 2: Example 2: Factors of 24
Problem: Find all factors of 24.
Solution:
24 = 1 × 24
24 = 2 × 12
24 = 3 × 8
24 = 4 × 6
Next: 5 — does 5 divide 24? No (24 ÷ 5 = 4 R 4). Stop.
Answer: Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24 (8 factors)
Example 3: Example 3: Factors of a Prime Number
Problem: Find all factors of 13.
Solution:
13 = 1 × 13
Try 2: 13 ÷ 2 = 6 R 1 (No)
Try 3: 13 ÷ 3 = 4 R 1 (No)
3 < √13 ≈ 3.6, so we stop.
Answer: Factors of 13 = 1, 13 (only 2 factors — 13 is a prime number).
Example 4: Example 4: Factors of 36 (Perfect Square)
Problem: Find all factors of 36.
Solution:
36 = 1 × 36
36 = 2 × 18
36 = 3 × 12
36 = 4 × 9
36 = 6 × 6
Answer: Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36 (9 factors — odd count because 6 × 6 gives one factor, not two)
Example 5: Example 5: Using Factor Pairs
Problem: Write all factor pairs of 20.
Solution:
| Factor Pair | Product |
|---|---|
| 1 × 20 | 20 |
| 2 × 10 | 20 |
| 4 × 5 | 20 |
Answer: Factor pairs: (1,20), (2,10), (4,5). Factors of 20 = 1, 2, 4, 5, 10, 20
Example 6: Example 6: Word Problem (Equal Rows)
Problem: Priya has 18 plants. She wants to arrange them in equal rows. What are the possible arrangements?
Solution:
Find factor pairs of 18:
1 × 18, 2 × 9, 3 × 6
Possible arrangements: 1 row of 18, 2 rows of 9, 3 rows of 6, 6 rows of 3, 9 rows of 2, or 18 rows of 1.
Answer: She can make 6 different arrangements.
Example 7: Example 7: Is 7 a Factor of 49?
Problem: Check if 7 is a factor of 49.
Solution:
49 ÷ 7 = 7 (exact division, remainder 0)
Answer: Yes, 7 is a factor of 49.
Example 8: Example 8: Factors of 1 and 0
Problem: (a) What are the factors of 1? (b) Is 0 a factor of any number?
Solution:
(a) 1 = 1 × 1. The only factor of 1 is 1.
(b) No. Division by 0 is undefined, so 0 is never a factor of any number. However, every number is a factor of 0 (since 0 ÷ any non-zero number = 0).
Example 9: Example 9: Number with Most Factors (up to 50)
Problem: Which number from 1 to 50 has the most factors?
Solution:
48 has factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 → 10 factors
36 has 9 factors. 40 has 8 factors. 48 wins.
Answer: 48 has the most factors (10 factors) among numbers 1 to 50.
Example 10: Example 10: Using Divisibility Rules to Find Factors
Problem: Find all factors of 30 using divisibility rules.
Solution:
30 ends in 0 → divisible by 2, 5, and 10.
Digit sum = 3 → divisible by 3.
Factor pairs: 1×30, 2×15, 3×10, 5×6.
Answer: Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30
Real-World Applications
Factors are used in many areas:
- Arranging objects: Finding how to arrange chairs in equal rows.
- Simplifying fractions: Dividing numerator and denominator by common factors.
- Equal distribution: Checking if items can be shared equally.
- Number theory: Finding HCF, LCM, prime factorisation.
Key Points to Remember
- A factor divides a number exactly (remainder = 0).
- 1 is a factor of every number. Every number is a factor of itself.
- Factors come in pairs that multiply to give the number.
- The total number of factors is always finite.
- A number with exactly 2 factors (1 and itself) is a prime number.
- A number with more than 2 factors is a composite number.
- Use divisibility rules to quickly identify factors.
Practice Problems
- Find all factors of 28.
- List the factor pairs of 40.
- Is 6 a factor of 54? How do you check?
- Find all factors of 50.
- Kavi has 32 chocolates. In how many ways can she arrange them in equal rows?
- Which is a prime number: 21 or 23? Find the factors of both to decide.
- Find all factors of 100.
- How many factors does 16 have? List them.
Frequently Asked Questions
Q1. What is the difference between factors and multiples?
Factors divide a number exactly (they are smaller or equal). Multiples are obtained by multiplying the number (they are larger or equal). For example, factors of 6 are 1,2,3,6 while multiples of 6 are 6,12,18,24,...
Q2. How many factors does a prime number have?
Exactly 2: the number 1 and the number itself. For example, 7 has only factors 1 and 7.
Q3. Is 1 a prime number?
No. A prime number must have exactly 2 factors. The number 1 has only 1 factor (itself), so it is neither prime nor composite.
Q4. How do I find factors quickly?
Start with 1 and the number. Then try 2, 3, 4, 5, etc. Use divisibility rules to skip numbers that clearly do not divide. Stop when the divisor exceeds the quotient.
Q5. Can a number have an odd number of factors?
Yes, but only perfect squares. For example, 25 has factors 1, 5, 25 (three factors) because 5 × 5 gives a repeated pair.
Q6. What are the factors of 0?
Every non-zero number is a factor of 0 because 0 divided by any non-zero number gives 0 (exact). So 0 has infinitely many factors.
Q7. How are factors related to division?
If a is a factor of n, then n ÷ a gives remainder 0. Factors are exactly the divisors that produce zero remainder.
Q8. Is finding factors covered in NCERT Class 4?
Yes. NCERT Class 4 Maths introduces factors through multiplication fact families and extends to factor pairs and identifying prime/composite numbers.
Related Topics
- Multiples of a Number
- Common Factors
- Common Multiples
- HCF (Highest Common Factor) Introduction
- LCM (Least Common Multiple) Introduction
- Prime and Composite Numbers
- Sieve of Eratosthenes
- Even and Odd Number Properties
- Factors and Multiples Word Problems
- Prime Numbers up to 100
- Factor Pairs
- Patterns in Multiples
