How prime factorization works
Prime factorization breaks a number into the prime numbers that multiply to give it. Divide repeatedly by the smallest prime that fits: 60 ÷ 2 = 30, ÷ 2 = 15, ÷ 3 = 5, leaving 5. So 60 = 2 × 2 × 3 × 5.
Always start with the smallest prime and keep dividing by it as long as it goes in evenly before moving up to the next one. Working in order guarantees no factor is missed. The result is often written with exponents: 60 = 2² × 3 × 5. Shared prime factors are exactly what the greatest common factor calculator collects.
Why prime factorization is unique
The Fundamental Theorem of Arithmetic guarantees every integer greater than 1 has exactly one prime factorization (ignoring order). This makes it the backbone of finding greatest common divisors, least common multiples and simplifying fractions — handy for the decimal to fraction calculator.
To use it in practice, factorise both numbers and compare. Taking the lowest power of each shared prime gives the greatest common divisor; taking the highest power of every prime that appears in either gives the least common multiple.
How to use this calculator
Enter a whole number of 2 or more. The result lists the prime factors as a product, like 2 × 2 × 3 × 5, plus a count of how many prime factors there are. Decimals and numbers below 2 are not valid inputs.
The number 1 is deliberately excluded from the primes: if it counted, you could pad any product with as many 1s as you liked and uniqueness would be lost. Take the highest power of each prime instead and you have the least common multiple.
Frequently asked questions
- What is the prime factorization of 60?
- 60 = 2 × 2 × 3 × 5. Multiplying those four primes gives 60.
- What counts as a prime number?
- A prime is a whole number greater than 1 with no divisors other than 1 and itself, such as 2, 3, 5 and 7.
- Why must the input be 2 or more?
- Prime factorization is only defined for integers greater than 1; 0 and 1 have no prime factors.
- Is the prime factorization always unique?
- Yes. Apart from the order of the factors, every integer above 1 has exactly one prime factorization.
- How do I know when to stop testing divisors?
- Stop once the test divisor exceeds the square root of the number that is left. If nothing has divided in by then, the remaining number is itself prime — for 60 you only need to test up to about 7.75.