Prime Factorisation
Every composite number can be broken down into a product of prime numbers. This process is called prime factorisation. Think of it as finding the building blocks — the smallest pieces — that a number is made of.
For example, 12 = 2 × 2 × 3. The prime numbers 2 and 3 are the building blocks of 12. No matter how you break 12 apart, you will always end up with the same prime factors: two 2s and one 3. You could start with 12 = 4 × 3 or 12 = 6 × 2 or 12 = 2 × 6, but when you break each factor down to primes, you always get 2 × 2 × 3.
In Class 6 NCERT Maths (Playing with Numbers), you will learn two methods to find the prime factorisation of a number: the factor tree method (where you keep splitting a number into factors until all factors are prime) and the division method (where you keep dividing by the smallest prime that works). Both methods give the same answer.
Prime factorisation is one of the most powerful tools in mathematics. It is used to find HCF and LCM, simplify fractions, check if a number is a perfect square or perfect cube, solve divisibility problems, and even secure your passwords on the internet (through cryptography).
Every composite number has a unique prime factorisation. This important fact is called the Fundamental Theorem of Arithmetic. It means that just as every building is made of a unique arrangement of bricks, every number has a unique arrangement of prime factors. No two different composite numbers have the same prime factorisation.
What is Prime Factorisation - Grade 6 Maths (Playing with Numbers)?
Definition: Prime factorisation is the process of expressing a composite number as a product of its prime factors.
Prime number: A number greater than 1 that has exactly two factors: 1 and itself.
- Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...
Composite number: A number greater than 1 that has more than two factors.
- Examples: 4, 6, 8, 9, 10, 12, 14, 15, ...
Prime factorisation:
- 12 = 2 × 2 × 3 = 2² × 3
- 30 = 2 × 3 × 5
- 100 = 2 × 2 × 5 × 5 = 2² × 5²
Important:
- 1 is neither prime nor composite. It is not included in prime factorisation.
- A prime number is its own prime factorisation (e.g., 7 = 7).
- The prime factorisation of a number is unique (except for the order of the factors).
Prime Factorisation Formula
Method 1: Factor Tree
- Write the number at the top.
- Split it into any two factors (other than 1 and itself if possible).
- If a factor is composite, split it further.
- Keep splitting until all factors are prime.
- The prime factors at the ends of all branches are the prime factorisation.
Method 2: Division Method (Repeated Division)
- Divide the number by the smallest prime that divides it.
- Write the quotient below.
- Divide the quotient by the smallest prime that divides it.
- Repeat until the quotient is 1.
- The prime factorisation is the product of all the divisors.
Writing in Index Form:
72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
The small raised number (exponent/index) tells how many times that prime factor appears.
Finding HCF using prime factorisation:
- Find the prime factorisation of each number.
- HCF = product of common prime factors with the smallest power.
Finding LCM using prime factorisation:
- Find the prime factorisation of each number.
- LCM = product of all prime factors with the highest power.
Types and Properties
Type 1: Prime Factorisation Using Factor Tree
- Start with any pair of factors. Keep breaking composites into smaller factors.
- Example: 36 → 6 × 6 → (2 × 3) × (2 × 3) → 2² × 3²
Type 2: Prime Factorisation Using Division Method
- Divide by 2, then 3, then 5, and so on until quotient is 1.
- Example: 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1. So 60 = 2² × 3 × 5.
Type 3: Writing in Index (Exponent) Form
- Group repeated primes and write with exponents.
- Example: 2 × 2 × 2 × 5 × 5 = 2³ × 5²
Type 4: Finding HCF Using Prime Factorisation
- Compare prime factorisations. Take common primes with lowest powers.
- Example: HCF of 12 (2² × 3) and 18 (2 × 3²) = 2¹ × 3¹ = 6
Type 5: Finding LCM Using Prime Factorisation
- Take all primes with highest powers.
- Example: LCM of 12 (2² × 3) and 18 (2 × 3²) = 2² × 3² = 36
Type 6: Checking if a Number is Prime
- Try dividing by primes up to the square root. If none divide it, it is prime.
- Example: Is 29 prime? √29 ≈ 5.4. Test 2, 3, 5. None divides 29. Yes, 29 is prime.
Solved Examples
Example 1: Factor Tree Method
Problem: Find the prime factorisation of 60 using a factor tree.
Solution:
Given:
- Number: 60
Steps:
- 60 = 6 × 10
- 6 = 2 × 3 (both prime — stop)
- 10 = 2 × 5 (both prime — stop)
- Collect all prime factors: 2, 3, 2, 5
- Arrange: 2 × 2 × 3 × 5 = 2² × 3 × 5
Answer: 60 = 2² × 3 × 5
Example 2: Division Method
Problem: Find the prime factorisation of 84 using the division method.
Solution:
Given:
- Number: 84
Steps:
- 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1
- Prime factors: 2 × 2 × 3 × 7
Answer: 84 = 2² × 3 × 7
Example 3: Large Number Factorisation
Problem: Find the prime factorisation of 360.
Solution:
Given:
- Number: 360
Steps:
- 360 ÷ 2 = 180
- 180 ÷ 2 = 90
- 90 ÷ 2 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
- Prime factors: 2 × 2 × 2 × 3 × 3 × 5
Answer: 360 = 2³ × 3² × 5
Example 4: Index Form
Problem: Write the prime factorisation of 108 in index form.
Solution:
Given:
- Number: 108
Steps:
- 108 ÷ 2 = 54
- 54 ÷ 2 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- Prime factors: 2 × 2 × 3 × 3 × 3
Answer: 108 = 2² × 3³
Example 5: HCF Using Prime Factorisation
Problem: Find the HCF of 36 and 48 using prime factorisation.
Solution:
Given:
- Numbers: 36 and 48
Steps:
- 36 = 2² × 3²
- 48 = 2⁴ × 3
- Common primes: 2 and 3
- Take the smallest power of each: 2² and 3¹
- HCF = 2² × 3 = 4 × 3 = 12
Answer: HCF of 36 and 48 = 12
Example 6: LCM Using Prime Factorisation
Problem: Find the LCM of 24 and 36 using prime factorisation.
Solution:
Given:
- Numbers: 24 and 36
Steps:
- 24 = 2³ × 3
- 36 = 2² × 3²
- Take all primes with highest power: 2³ and 3²
- LCM = 2³ × 3² = 8 × 9 = 72
Answer: LCM of 24 and 36 = 72
Example 7: Checking if a Number is Prime
Problem: Is 53 a prime number?
Solution:
Given:
- Number: 53
Steps:
- √53 ≈ 7.3. Test primes up to 7: 2, 3, 5, 7.
- 53 ÷ 2 = 26.5 (not exact)
- 53 ÷ 3 = 17.67 (not exact)
- 53 ÷ 5 = 10.6 (not exact)
- 53 ÷ 7 = 7.57 (not exact)
- No prime up to 7 divides 53.
Answer: Yes, 53 is a prime number.
Example 8: Word Problem — Arranging Students
Problem: A school has 120 students. The principal wants to arrange them in equal rows for an assembly. What are all the possible row sizes that use only prime numbers of students per row?
Solution:
Given:
- Total students: 120
Steps:
- Prime factorisation: 120 = 2³ × 3 × 5
- Prime factors of 120 are: 2, 3, 5
- Rows of 2 students: 120 ÷ 2 = 60 rows
- Rows of 3 students: 120 ÷ 3 = 40 rows
- Rows of 5 students: 120 ÷ 5 = 24 rows
Answer: Possible prime row sizes: 2, 3, or 5 students per row.
Example 9: Factor Tree (Different Starting Point)
Problem: Find the prime factorisation of 72 using two different factor trees. Do you get the same answer?
Solution:
Given:
- Number: 72
Steps (Tree 1):
- 72 = 8 × 9
- 8 = 2 × 4 = 2 × 2 × 2
- 9 = 3 × 3
- Result: 2 × 2 × 2 × 3 × 3 = 2³ × 3²
Steps (Tree 2):
- 72 = 12 × 6
- 12 = 2 × 6 = 2 × 2 × 3
- 6 = 2 × 3
- Result: 2 × 2 × 3 × 2 × 3 = 2³ × 3²
Answer: Both trees give 72 = 2³ × 3². The prime factorisation is always the same.
Example 10: Perfect Square Check
Problem: Is 144 a perfect square? Use prime factorisation to check.
Solution:
Given:
- Number: 144
Steps:
- 144 = 2 × 72 = 2 × 2 × 36 = 2 × 2 × 2 × 18 = 2 × 2 × 2 × 2 × 9 = 2 × 2 × 2 × 2 × 3 × 3
- 144 = 2⁴ × 3²
- All exponents (4 and 2) are even.
- When all exponents in the prime factorisation are even, the number is a perfect square.
- 144 = (2² × 3)² = 12²
Answer: Yes, 144 is a perfect square (12² = 144).
Real-World Applications
Real-world uses of prime factorisation:
- Finding HCF and LCM: The most direct and common use. Prime factorisation makes finding HCF (common prime factors with smallest powers) and LCM (all prime factors with highest powers) systematic, reliable, and efficient — especially for large numbers where listing all factors would be impractical. This is why prime factorisation is taught before HCF and LCM.
- Simplifying Fractions: To simplify 12/18, find the HCF of 12 and 18 using prime factorisation: 12 = 2² × 3, 18 = 2 × 3². HCF = 2 × 3 = 6. So 12/18 = 2/3. Every fraction simplification ultimately relies on finding common prime factors of the numerator and denominator.
- Sharing and Distribution: If you have 90 sweets and want to share them equally into bags, the possible bag sizes are the factors of 90. Prime factorisation (90 = 2 × 3² × 5) helps you find all factors systematically: from the prime factorisation, you know the factors are combinations of 2⁰ or 2¹, times 3⁰ or 3¹ or 3², times 5⁰ or 5¹, giving 2 × 3 × 2 = 12 factors total.
- Cryptography and Internet Security: This is perhaps the most surprising application. Internet security, online banking, and digital payments all use very large prime numbers (hundreds of digits long). The RSA encryption algorithm relies on the fact that multiplying two large primes is easy, but factorising the product back into those primes is extremely hard. Every time you shop online or log in to a website, prime factorisation (or rather, the difficulty of it) is keeping your data safe.
- Perfect Squares and Cubes: Prime factorisation immediately tells you if a number is a perfect square (all exponents are even) or a perfect cube (all exponents are divisible by 3). For example, 144 = 2⁴ × 3² — all exponents (4, 2) are even, so 144 is a perfect square (12²). And 216 = 2³ × 3³ — all exponents (3, 3) are divisible by 3, so 216 is a perfect cube (6³).
- Adding Fractions (LCD): When adding fractions with different denominators (e.g., 1/12 + 1/18), you need the Least Common Denominator, which is the LCM of the denominators. Prime factorisation gives: LCM of 12 and 18 = 2² × 3² = 36. So you convert to thirty-sixths: 3/36 + 2/36 = 5/36.
- Music Theory: Musical harmony is based on ratios of small numbers. The octave is a 2:1 ratio, the perfect fifth is 3:2, the perfect fourth is 4:3. These simple ratios involve small primes (2 and 3). Understanding why certain intervals sound harmonious connects back to the prime factors of frequency ratios.
Key Points to Remember
- Prime factorisation expresses a composite number as a product of prime numbers. Example: 60 = 2² × 3 × 5.
- Every composite number has a unique prime factorisation — this is the Fundamental Theorem of Arithmetic. No two different numbers share the same prime factorisation.
- Two methods to find it: Factor Tree (split into factors, keep splitting composites) and Division Method (divide by smallest prime repeatedly until quotient is 1). Both always give the same result.
- 1 is neither prime nor composite and is never included in any prime factorisation.
- In index (exponent) form, repeated primes are written with a superscript: 2 × 2 × 2 = 2³ ("two cubed" or "two to the power 3").
- HCF using prime factorisation: Take common prime factors with smallest powers. HCF of 24 (2³ × 3) and 36 (2² × 3²) = 2² × 3 = 12.
- LCM using prime factorisation: Take all prime factors with highest powers. LCM of 24 (2³ × 3) and 36 (2² × 3²) = 2³ × 3² = 72.
- To check if a number is prime: test divisibility by all primes up to its square root. If none divide it, it is prime.
- A number is a perfect square if all exponents in its prime factorisation are even. Example: 144 = 2⁴ × 3² (exponents 4, 2 are even, so 144 = 12²).
- A number is a perfect cube if all exponents are divisible by 3. Example: 216 = 2³ × 3³ (both divisible by 3, so 216 = 6³).
- HCF × LCM = Product of the two numbers. For 24 and 36: HCF × LCM = 12 × 72 = 864 = 24 × 36.
- Different factor trees for the same number always give the same prime factorisation (just in different order).
Practice Problems
- Find the prime factorisation of 90 using the factor tree method.
- Find the prime factorisation of 150 using the division method.
- Write the prime factorisation of 200 in index form.
- Find the HCF of 42 and 70 using prime factorisation.
- Find the LCM of 16 and 20 using prime factorisation.
- Is 97 a prime number? Show your working.
- The prime factorisation of a number is 2³ × 5² × 7. What is the number?
- Is 225 a perfect square? Use prime factorisation to check.
Frequently Asked Questions
Q1. What is prime factorisation?
Prime factorisation is breaking a composite number into a product of prime numbers. For example, 30 = 2 × 3 × 5. Every composite number can be written this way, and the result is unique.
Q2. What are the two methods of prime factorisation?
The factor tree method (split the number into any two factors, keep splitting composites) and the division method (divide by the smallest prime repeatedly until the quotient is 1).
Q3. Can 1 be a prime factor?
No. 1 is not a prime number (it has only one factor, not two). Prime factorisation uses only prime numbers: 2, 3, 5, 7, 11, etc.
Q4. Can two different factor trees give different prime factorisations?
No. You may start with different pairs of factors, but the final set of prime factors is always the same. For example, 24 = 4 × 6 or 24 = 3 × 8 both lead to 2³ × 3.
Q5. How does prime factorisation help find HCF?
Write the prime factorisation of each number. The HCF is the product of the common prime factors, each raised to the smallest power it appears with. Example: HCF of 12 (2² × 3) and 18 (2 × 3²) = 2¹ × 3¹ = 6.
Q6. How does prime factorisation help find LCM?
Write the prime factorisation of each number. The LCM is the product of all prime factors (from both numbers), each raised to the highest power it appears with. Example: LCM of 12 (2² × 3) and 18 (2 × 3²) = 2² × 3² = 36.
Q7. What is the Fundamental Theorem of Arithmetic?
It states that every composite number can be expressed as a product of prime numbers in exactly one way (except for the order of the factors). This means prime factorisation is unique for every number.
Q8. How do you check if a large number is prime?
Find the square root of the number. Test divisibility by all prime numbers up to that square root. If none of them divides the number exactly, it is prime. For example, to check 101: √101 ≈ 10. Test 2, 3, 5, 7. None divides 101. So 101 is prime.
