Factor Calculator: All Factors & Prime Factorization
Free factor calculator: find all factors of any integer, prime factorization, and factor tree. How to find all factors of a number, list factors of 60, prime factorization calculator. n mod a = 0 up to โn. Teaching and RSA context.
What This Calculator Does & Who It's For
Calculator Purpose & Ideal Users
- What You'll Get:All factors: Every integer a such that n mod a = 0, listed in order (e.g. list factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60). Prime factorization: The number as a product of primes (e.g. 2ยณ ร 3 ร 5). Factor tree: An SVG diagram showing how the number splits into factors until primes. Step-by-step: A short explanation of the division steps.
- Ideal Users:Students & teachers: Number theory, what are factors in math, primes, factor trees. Homework: "Find all factors of 60" or "Write the prime factorization." Curious readers: RSA and why factoring large numbers is hard. Use as a prime factorization calculator or to find all factors of a number.
- Scope & Limits:Positive integers only, up to 12 digits. Trial division up to โn; no imaginary or decimal inputs.
This factor calculator finds every factor of a single positive integer and its prime factorizationโideal when you need to list all factors or get a prime factorization for homework or teaching.
What Is a Factor? Definition and n mod a = 0
A factor of an integer n is a positive integer a that divides n evenlyโi.e. the remainder is zero. In other words:
n mod a = 0
(or n = a ร k for some integer k). For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. To find all factors of a number, it is enough to check a from 1 up to โn: whenever n mod a = 0, both a and n รท a are factors. This factor calculator uses that method for integers up to 12 digits.Prime Numbers and Prime Factorization
A prime number is an integer greater than 1 whose only positive divisors are 1 and itself (e.g. 2, 3, 5, 7). Prime factorization is writing a number as a product of primes raised to powers, e.g.
60 = 2ยฒ ร 3ยน ร 5ยน
(the prime factorization of 60). By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization (up to the order of factors). This prime factorization calculator finds that factorization using trial division: divide by the smallest prime repeatedly until the quotient is 1 or prime.The RSA Challenge: Why Factoring Is Hard
In RSA and similar cryptosystems, security relies on the fact that multiplying two large primes is easy, but factoring their product back into primes is computationally very hard. So the "one-way" nature of factoring is what makes the system secure. In practice, RSA uses numbers with hundreds of digits; even the best known algorithms cannot factor them in reasonable time. This factor calculator handles integers up to 12 digitsโenough for teaching factors, prime factorization, and factor treesโand illustrates the same trial-division idea that underpins understanding of the RSA challenge.
Factor Calculator FAQ
? What is a factor of a number?
A factor of an integer n is any positive integer a such that n is divisible by a with no remainderโi.e. n mod a = 0. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. This factor calculator finds every factor of a single number up to 12 digits using trial division up to โn.
? What is prime factorization?
Prime factorization is writing a number as a product of prime numbers raised to powers, e.g. 60 = 2ยฒ ร 3 ร 5. Every integer greater than 1 has a unique prime factorization (up to order). This calculator shows the prime factorization and a factor tree so you can see how the number breaks down into primes.
? How do you find all factors of a number?
To find every factor of n, check all integers a from 1 up to โn: if n mod a = 0, then both a and n รท a are factors. That way you only need to loop up to โn instead of n. This factor calculator uses that method and lists factors in ascending order.
? Why is prime factorization important? (RSA and cryptography)
Prime factorization is hard for very large numbers: multiplying two large primes is easy, but factoring their product back into primes is computationally difficult. That one-way asymmetry is the basis of RSA and other public-key cryptosystems. In practice, RSA uses numbers so large that even the best algorithms cannot factor them in reasonable time; this calculator handles integers up to 12 digits for teaching and small-number use.
? What is the maximum number I can enter?
This calculator accepts positive integers up to 12 digits (999,999,999,999). It uses trial division up to the square root of the number, so larger inputs would slow down. For arbitrary-precision factoring or very large numbers, use a dedicated big-number or prime-factorization tool.