Prime Factorisation
Prime factorisation is the process of expressing a composite number as a product of its prime factors. Every composite number can be broken down into prime numbers, and this breakdown is unique (except for the order of the factors).
For example, 60 = 2 x 2 x 3 x 5. No matter how you break 60 down — whether you start by splitting it into 6 x 10 or 4 x 15 — you will always get the same prime factors: two 2s, one 3, and one 5.
Prime factorisation is a key skill in Class 5 because it is the basis for finding the HCF and LCM of two or more numbers, simplifying fractions, and solving divisibility problems.
Before learning prime factorisation, students should be confident with:
- Identifying prime numbers (2, 3, 5, 7, 11, 13, ...)
- Identifying composite numbers (4, 6, 8, 9, 10, 12, ...)
- Divisibility rules for 2, 3, 4, 5, 6, 9, and 10
What is Prime Factorisation - Class 5 Maths (Factors and Multiples)?
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, 29.
Composite number: A number greater than 1 that has more than two factors. Examples: 4, 6, 8, 9, 10, 12, 14, 15.
Note: 1 is neither prime nor composite.
Prime factorisation: Expressing a composite number as a product of prime numbers only.
| Number | Prime Factorisation |
|---|---|
| 12 | 2 x 2 x 3 |
| 36 | 2 x 2 x 3 x 3 |
| 60 | 2 x 2 x 3 x 5 |
| 100 | 2 x 2 x 5 x 5 |
Prime Factorisation Formula
Every composite number = Product of prime factors (unique)
This is called the Fundamental Theorem of Arithmetic. No matter which method you use (factor tree or repeated division), the prime factors are always the same.
Types and Properties
Two methods of prime factorisation:
Method 1: Factor Tree
- Write the number at the top.
- Break it into any two factors (not 1 and the number).
- If a factor is prime, circle it (it is done).
- If a factor is composite, break it further.
- Continue until all branches end in prime numbers.
- Multiply all circled primes — that is the prime factorisation.
Example: Factor tree for 72
72
/ \
8 9
/ \ / \
2 4 3 3
/ \
2 2
72 = 2 x 2 x 2 x 3 x 3
Method 2: Repeated Division (Ladder Method)
- Divide the number by the smallest prime factor (start with 2).
- Write the quotient below.
- Divide the quotient by the smallest prime factor again.
- Continue until the quotient is 1.
- The prime factorisation is the product of all the divisors.
Example: Repeated division for 72
| 2 | 72 |
| 2 | 36 |
| 2 | 18 |
| 3 | 9 |
| 3 | 3 |
| 1 |
72 = 2 x 2 x 2 x 3 x 3
Solved Examples
Example 1: Example 1: Prime Factorisation of 36
Problem: Find the prime factorisation of 36.
Solution (Repeated Division):
36 / 2 = 18
18 / 2 = 9
9 / 3 = 3
3 / 3 = 1
Answer: 36 = 2 x 2 x 3 x 3
Example 2: Example 2: Prime Factorisation of 84
Problem: Express 84 as a product of prime factors.
Solution (Repeated Division):
84 / 2 = 42
42 / 2 = 21
21 / 3 = 7
7 / 7 = 1
Answer: 84 = 2 x 2 x 3 x 7
Example 3: Example 3: Factor Tree for 120
Problem: Draw a factor tree for 120 and write its prime factorisation.
Solution:
120
/ \
10 12
/ \ / \
2 5 4 3
/ \
2 2
Answer: 120 = 2 x 2 x 2 x 3 x 5
Example 4: Example 4: Prime Factorisation of 200
Problem: Find the prime factorisation of 200.
Solution (Repeated Division):
200 / 2 = 100
100 / 2 = 50
50 / 2 = 25
25 / 5 = 5
5 / 5 = 1
Answer: 200 = 2 x 2 x 2 x 5 x 5
Example 5: Example 5: Prime Factorisation of a 3-Digit Number
Problem: Express 360 as a product of prime factors.
Solution (Repeated Division):
360 / 2 = 180
180 / 2 = 90
90 / 2 = 45
45 / 3 = 15
15 / 3 = 5
5 / 5 = 1
Answer: 360 = 2 x 2 x 2 x 3 x 3 x 5
Example 6: Example 6: Is the Number Prime?
Problem: Is 97 a prime number? Use prime factorisation to check.
Solution:
Try dividing 97 by all primes up to its square root (approximately 9.8):
- 97 / 2 → not exact (97 is odd)
- 97 / 3 → not exact (9 + 7 = 16, not divisible by 3)
- 97 / 5 → not exact (doesn't end in 0 or 5)
- 97 / 7 = 13.8... → not exact
No prime divides 97 exactly.
Answer: 97 is a prime number.
Example 7: Example 7: Expressing with Exponents
Problem: Write the prime factorisation of 180 using exponents.
Solution:
180 / 2 = 90, 90 / 2 = 45, 45 / 3 = 15, 15 / 3 = 5, 5 / 5 = 1
180 = 2 x 2 x 3 x 3 x 5
Using exponents: 180 = 2² x 3² x 5
Answer: 180 = 2² x 3² x 5
Example 8: Example 8: Word Problem — Arranging Students
Problem: A teacher has 90 students. She wants to arrange them in equal rows with the same number of students in each row. Using prime factorisation, find all possible arrangements.
Solution:
Step 1: 90 = 2 x 3 x 3 x 5
Step 2: Factors of 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Step 3: Each factor gives an arrangement (rows x students per row):
- 1 row of 90, 2 rows of 45, 3 rows of 30, 5 rows of 18
- 6 rows of 15, 9 rows of 10, 10 rows of 9, etc.
Answer: There are 12 possible arrangements.
Example 9: Example 9: Different Factor Trees, Same Result
Problem: Show that two different factor trees of 60 give the same prime factorisation.
Solution:
Tree 1: 60 → 6 x 10 → (2 x 3) x (2 x 5) = 2 x 2 x 3 x 5
Tree 2: 60 → 4 x 15 → (2 x 2) x (3 x 5) = 2 x 2 x 3 x 5
Answer: Both trees give 60 = 2 x 2 x 3 x 5. The prime factorisation is unique.
Example 10: Example 10: Finding a Number from Prime Factors
Problem: A number has the prime factorisation 2 x 2 x 3 x 5 x 7. What is the number?
Solution:
Step 1: Multiply: 2 x 2 = 4
Step 2: 4 x 3 = 12
Step 3: 12 x 5 = 60
Step 4: 60 x 7 = 420
Answer: The number is 420.
Real-World Applications
Where prime factorisation is used:
- Finding HCF: Write the prime factorisation of each number. Take the common prime factors with the lowest powers. Multiply them to get the HCF.
- Finding LCM: Write the prime factorisation of each number. Take all prime factors with the highest powers. Multiply them to get the LCM.
- Simplifying fractions: Find common prime factors of the numerator and denominator. Divide both by these to reduce the fraction to its simplest form. Example: 18/24 → both have factors 2 and 3 → 18/24 = 3/4.
- Checking if a number is a perfect square: A number is a perfect square if every prime factor appears an even number of times. 144 = 2 x 2 x 2 x 2 x 3 x 3 (all even) → perfect square. 72 = 2 x 2 x 2 x 3 x 3 (three 2s, which is odd) → not a perfect square.
- Finding the smallest perfect square multiple: For 12 = 2² x 3, the factor 3 appears once (odd). Multiply by 3 to make it even: 12 x 3 = 36 = 2² x 3², which is a perfect square.
- Cryptography (advanced): In higher maths, the difficulty of factorising very large numbers into primes is the basis of internet security (RSA encryption).
Tips for quick prime factorisation:
- Always start dividing by the smallest prime (2). If the number is even, keep dividing by 2.
- Move to 3 when the number is no longer divisible by 2.
- Then try 5, 7, 11, and so on.
- Use divisibility rules to check quickly: even number → divisible by 2; digit sum divisible by 3 → divisible by 3; ends in 0 or 5 → divisible by 5.
Key Points to Remember
- A prime number has exactly 2 factors: 1 and itself.
- A composite number has more than 2 factors.
- 1 is neither prime nor composite.
- 2 is the only even prime number.
- Every composite number can be written as a unique product of primes (Fundamental Theorem of Arithmetic).
- Use the factor tree or repeated division (ladder) method to find prime factors.
- Always start dividing by the smallest prime (2, then 3, then 5, ...).
- To check if a number is prime, try dividing by all primes up to its square root.
Practice Problems
- Find the prime factorisation of 56.
- Express 144 as a product of prime factors.
- Draw a factor tree for 90.
- Write the prime factorisation of 252 using exponents.
- Is 87 a prime number? Check using prime factorisation.
- Find a number whose prime factorisation is 2 x 3 x 3 x 11.
- Find the prime factorisation of 500.
- Priya says 150 = 2 x 3 x 5 x 5. Ria says 150 = 2 x 5 x 15. Who is correct and why?
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 x 3 x 5. Every composite number has a unique set of prime factors.
Q2. What are the two methods of prime factorisation?
The two methods are the factor tree method (split the number into any two factors, then keep splitting composites) and the repeated division (ladder) method (keep dividing by the smallest prime until you reach 1). Both methods give the same prime factors.
Q3. Why is 1 not a prime number?
A prime number must have exactly two distinct factors: 1 and itself. The number 1 has only one factor (itself), so it does not meet the definition. Including 1 as prime would also break the uniqueness of prime factorisation.
Q4. Why is 2 the only even prime number?
Every even number greater than 2 is divisible by 2, so it has at least three factors (1, 2, and itself). Since prime numbers have exactly two factors, 2 is the only even number that qualifies.
Q5. Does the factor tree method always give the same answer?
Yes. Different factor trees may split the number differently at each step, but the final set of prime factors is always the same. For example, 24 can start as 4 x 6 or 3 x 8, but both lead to 2 x 2 x 2 x 3.
Q6. How do you know when to stop in repeated division?
Stop when the quotient becomes 1. At that point, all the divisors you used are the prime factors. For example, for 30: 30/2=15, 15/3=5, 5/5=1. Stop. Prime factorisation = 2 x 3 x 5.
Q7. What is the Fundamental Theorem of Arithmetic?
It states that every whole number greater than 1 can be expressed as a product of primes in exactly one way (ignoring the order). For example, 42 = 2 x 3 x 7 is the only way to write 42 as a product of primes.
Q8. How is prime factorisation used to check for perfect squares?
A number is a perfect square if every prime factor appears an even number of times. For example, 36 = 2 x 2 x 3 x 3 (each appears twice), so 36 is a perfect square. But 48 = 2 x 2 x 2 x 2 x 3 (3 appears once), so 48 is not a perfect square.
Q9. Can prime factorisation be used for large numbers?
Yes, but it gets harder as numbers grow. For Class 5, prime factorisation is typically done for numbers up to a few hundred. For very large numbers (millions and beyond), computers use advanced algorithms.
Q10. What is the difference between factors and prime factors?
Factors include all numbers that divide a number exactly — both prime and composite. Prime factors are only the prime numbers in that list. For 12: factors are 1, 2, 3, 4, 6, 12, but prime factors are only 2 and 3.
