Break any positive integer into its prime factors, displayed as a factor tree and in exponential notation.
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Number Theory
A prime factorisation calculator decomposes any positive integer greater than one into a product of prime numbers. The result is unique for every integer, a property known as the fundamental theorem of arithmetic, and is used in GCF and LCM calculations, cryptography, and simplifying radicals.
Every integer greater than one can be written as a product of prime numbers in exactly one way, ignoring the order of the factors. This unique decomposition reveals the building blocks of a number and is the foundation of many results in number theory.
For example, 60 equals 2 times 2 times 3 times 5, often written as two squared times three times five. No other combination of primes produces 60.
Start by dividing the number by the smallest prime, two. If it divides evenly, record two as a factor and divide. Repeat until two no longer divides evenly, then move to the next prime, three, and continue. The process ends when the remaining quotient is one.
n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ
The number n is expressed as a product of prime powers, where each pᵢ is a distinct prime and aᵢ is the number of times it appears.
Prime factorisation is used to compute the GCF and LCM of two numbers, simplify square roots and other radicals, determine whether a number is a perfect square or cube, and in cryptographic algorithms that rely on the difficulty of factoring very large numbers.
Frequently asked questions
No. By convention, one is neither prime nor composite. Prime factorisation applies only to integers greater than one.
Yes. The LCM of two numbers is the product of the highest power of each prime that appears in either factorisation. This is often faster than listing multiples.
A factor tree is a visual diagram that breaks a number into two factors at each branch until every leaf is a prime. The primes at the leaves are the prime factorisation of the original number.
Related
These related calculators come from the same leaf category, nearby sibling categories, or the same top-level topic.
Find all factors of any positive integer, displayed as individual factors and factor pairs, with a prime-or-composite check.
Find the greatest common factor of two or more numbers using prime factorisation, with step-by-step working and factor lists.
Find the least common multiple or greatest common divisor of two whole numbers with step-by-step working.
Perform arithmetic in any number base from 2 to 36 and view results across binary, octal, decimal, and hexadecimal.
