Direct answer
Every integer greater than one has a prime factorisation that is unique apart from factor order; repeated exact division reveals the prime powers composing the number.
What this calculation tells you
Prime factorisation rewrites a positive integer as a product of primes and their exponents. It exposes divisor structure and supports GCF, LCM, fraction simplification, and modular reasoning.
The factorisation of zero is not defined in the same way because every nonzero integer divides zero; one has an empty prime product.
Where it is used
Arithmetic and education
Simplify fractions and explain divisibility.
Number theory
Study divisors, congruences, and multiplicative functions.
Computing
Understand why large-integer factorisation matters in some cryptographic systems without treating a classroom tool as security software.
Scheduling and grouping
Support common-cycle and equal-group calculations through GCF and LCM.
When this guide helps
- Factoring a composite integer.
- Checking whether a number is prime.
- Building GCF and LCM from exponents.
- Counting or summing divisors.
Divide by primes systematically
Start with the smallest possible prime and divide repeatedly while exact, then continue until the remaining quotient is one or prime.
Read exponents as multiplicity
The exponent records how many times a prime occurs. These exponent schedules make divisor and common-factor relationships transparent.
Respect special integers
Negative integers add a sign to the factorisation of their magnitude. Zero and one require explicit conventions rather than ordinary prime lists.
Common mistakes
Frequent errors include stopping at composite factors and calling one prime.
- Test every remaining factor.
- Verify by multiplication.
- State the domain and sign convention.
Worked case: factor 360
Find the prime-power form of 360.
Divide successively: 360=36x10=(2²x3²)(2x5)=2³x3²x5.
The prime factorisation is 2³ x 3² x 5.
Multiplying 8x9x5 returns 360.
Reproduce this worked caseOpen Prime Factorization Calculator
Worked case: use factors to count divisors
Use 360=2³x3²x5¹ to count positive divisors.
Each divisor chooses exponents 0–3, 0–2 and 0–1, giving (3+1)(2+1)(1+1)=24.
360 has 24 positive divisors.
This companion result follows from the verified factorisation and should not be guessed from magnitude.
Reproduce this worked caseOpen Prime Factorization Calculator
Compare prime factorisation cases before generalising
One is neither prime nor composite and has an empty prime product convention. Negative integers require a separate sign followed by factorisation of their magnitude.
| Integer | Prime powers | Positive divisor count |
|---|---|---|
| 360 | 2³x3²x5 | 24 |
| 84 | 2²x3x7 | 12 |
prime factorisation: calculation checklist
- Handle sign separately
- Use prime factors only
- Combine repeated factors as powers
- Multiply back
- Treat 1 explicitly
Practical questions
Frequently asked questions
Is 1 a prime number?
No. A prime has exactly two positive divisors; one has only one.
Why is factorisation unique?
The fundamental theorem of arithmetic guarantees one prime-power structure for each integer greater than one, apart from ordering.
Can zero be prime-factorised?
No finite prime product uniquely represents zero.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.