Prime Factorisation Calculator

Enter a whole number (2–10,000,000)

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.

How the Prime Factorisation Calculator Works

This calculator helps you understand your financial position using current UK rates and regulations for the 2025/26 tax year. Whether you are planning savings, evaluating loan options, or projecting investment growth, accurate calculations are essential for making informed decisions about your money.

UK financial products are regulated by the Financial Conduct Authority (FCA). Interest rates, fees, and terms vary significantly between providers, so comparing actual costs rather than headline rates is important. This tool gives you a clear picture to inform your comparisons.

Key Information for 2025/26

The Bank of England base rate is 4.5% as of early 2026. The Personal Savings Allowance lets basic rate taxpayers earn up to £1,000 in savings interest tax-free (£500 for higher rate taxpayers). The annual ISA allowance remains at £20,000, and the Lifetime ISA allowance is £4,000 with a 25% government bonus for first-time buyers or retirement savings.

Example Calculation

Saving £200 per month into an account earning 4.5% AER would grow to approximately £2,454 after one year, including £54 in interest. Over 5 years at the same rate, your £12,000 in contributions would grow to roughly £13,362, earning £1,362 in compound interest.

Source: Based on current UK financial rates. Last updated March 2026.

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.