Prime Factorisation Calculator

Q: What is prime factorisation?

Prime factorisation expresses a number as a product of prime numbers. Every whole number greater than 1 can be written uniquely as a product of primes (Fundamental Theorem of Arithmetic). For example, 360 = 2³ × 3² × 5.

Q: Is 1 a prime number?

No, 1 is not a prime number. A prime number must have exactly two distinct factors: 1 and itself. The number 1 only has one factor (itself), so it does not meet the definition. The smallest prime is 2.

Q: How do you find the HCF using prime factors?

Write both numbers as products of prime factors. Identify the prime factors they share. Multiply the shared primes together (using the lower power where they differ). Example: HCF(12,18): 12=2²×3, 18=2×3². Shared: 2¹×3¹=6.

Q: How do you find the LCM using prime factors?

Write both numbers as products of prime factors. Take each prime factor that appears in either number, using the highest power. Multiply them. Example: LCM(12,18): 12=2²×3, 18=2×3². Take 2²×3²=4×9=36.

Q: What is the factor tree method?

A factor tree splits a number into two factors, then splits each factor further until all branches end in prime numbers. For 360: 360=36×10, then 36=6×6=2×3×2×3, and 10=2×5. Collecting all primes: 2³×3²×5.

Q: Why is prime factorisation used in cryptography?

The RSA encryption algorithm relies on the fact that multiplying two large prime numbers together is easy, but factoring the result back into the two primes is computationally very hard. This one-way difficulty is the basis of internet security (HTTPS, TLS).

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HCF and LCM Calculator

Enter two numbers to find their Highest Common Factor and Lowest Common Multiple using prime factorisation.

First Number

Second Number

What Is Prime Factorisation?

Prime factorisation (also called prime decomposition) expresses a composite number as a product of prime numbers. The Fundamental Theorem of Arithmetic guarantees that every integer greater than 1 has exactly one prime factorisation (ignoring order).

Example — Factorise 360:
360 ÷ 2 = 180
180 ÷ 2 = 90
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
5 is prime — stop
360 = 2³ × 3² × 5

In index (powers) notation this is written as 2³ × 3² × 5. This tells us everything about the number's divisibility.

Prime Numbers — The Building Blocks

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. It cannot be formed by multiplying two smaller natural numbers together.

The first 50 prime numbers:

Is 1 prime? No. The number 1 has only one factor (itself), not two. It is classified as a unit, separate from primes and composites.

Is 2 prime? Yes — it is the only even prime number. All other even numbers are divisible by 2, so they have at least three factors.

Largest known prime (as of 2024): 2^136,279,841 − 1 (a Mersenne prime with over 41 million digits, discovered in October 2024).

Finding HCF Using Prime Factors

To find the Highest Common Factor of two or more numbers using prime factorisation:

Write each number as a product of prime factors
Identify the prime factors that appear in all numbers
For each shared prime, take the lower power
Multiply these together — the result is the HCF

Example — HCF(60, 72):
60 = 2² × 3 × 5
72 = 2³ × 3²
Shared primes: 2 and 3
Lower powers: 2² and 3¹
HCF = 2² × 3 = 4 × 3 = 12

Finding LCM Using Prime Factors

To find the Lowest Common Multiple using prime factorisation:

Write each number as a product of prime factors
Take each prime factor that appears in any of the numbers
For each prime, take the highest power seen
Multiply these together — the result is the LCM

Example — LCM(60, 72):
60 = 2² × 3 × 5
72 = 2³ × 3²
All primes: 2, 3, 5
Highest powers: 2³, 3², 5¹
LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360

Why Prime Factorisation Matters

Simplifying fractions: Divide numerator and denominator by their HCF. To find HCF quickly, use prime factorisation.
Finding LCM: Essential when adding fractions with different denominators (finding the lowest common denominator).
Cryptography (RSA encryption): Your HTTPS web browsing is secured by RSA, which relies on the difficulty of factoring large numbers (e.g. 2048-bit primes).
GCSE and A-level maths: Prime factorisation appears in number theory, algebraic simplification, and solving problems involving HCF and LCM.
Computer science: Hash functions, checksums, and random number generation use properties of prime numbers.

Frequently Asked Questions

What is prime factorisation?

Prime factorisation expresses any whole number greater than 1 as a unique product of prime numbers. For example, 360 = 2³ × 3² × 5. This is guaranteed by the Fundamental Theorem of Arithmetic — every integer has exactly one prime factorisation (ignoring the order of factors).

Is 1 a prime number?

No. A prime number must have exactly two distinct factors: 1 and itself. The number 1 only has one factor (1 itself), so it does not qualify. Classifying 1 as prime would break the uniqueness of prime factorisation (e.g. 6 = 2×3 = 1×2×3 = 1×1×2×3...). The smallest prime number is 2.

How do you find the HCF using prime factors?

Factorise both numbers. Identify shared prime factors and take the lower power of each. Multiply. For HCF(60,72): 60=2²×3×5, 72=2³×3². Shared primes: 2 (lower power=2²) and 3 (lower power=3¹). HCF = 4×3 = 12. Check: 12 divides 60 (=5) and 72 (=6). ✓

How do you find the LCM using prime factors?

Factorise both numbers. List all primes that appear in either. Take the highest power of each. Multiply. For LCM(60,72): 60=2²×3×5, 72=2³×3². All primes: 2 (highest=2³), 3 (highest=3²), 5 (highest=5¹). LCM = 8×9×5 = 360. Check: 360÷60=6 and 360÷72=5. ✓

What is the factor tree method?

A factor tree splits a number into any two factors, then splits each factor further until all branches end at prime numbers. For 60: split into 6×10, then 6=2×3 and 10=2×5. Primes collected: 2, 3, 2, 5 = 2²×3×5. Unlike short division, the factor tree is more visual and flexible — any split works as long as you reach all primes.

Why is prime factorisation used in cryptography?

RSA encryption uses two very large prime numbers (each hundreds of digits long). Their product forms the public key — easy to compute but computationally infeasible to reverse-factorise. Factoring a 2048-bit RSA number would take the world's fastest supercomputers longer than the age of the universe. This computational asymmetry secures internet banking, email, and HTTPS.