Prime Factorization Calculator
Decompose any positive integer into its prime factors. Displays the result as a product of prime powers and includes a factor tree visualization.
About this Calculator
Decompose any positive integer into its prime factors. Displays the result as a product of prime powers and includes a factor tree visualization.
Formula & Calculations
Formula
n = p1^e1 × p2^e2 × ... × pk^ek, where each pi is primeWhere:
- n=A positive integer greater than 1 to be factorized
- p1, p2, ..., pk=The distinct prime factors of n
- e1, e2, ..., ek=The exponent (multiplicity) of each prime factor
Assumptions
- Every integer greater than 1 has a unique prime factorization (Fundamental Theorem of Arithmetic).
- Trial division up to sqrt(n) is used, which is efficient for moderate-sized numbers.
- The number 1 has no prime factors (empty product).
Calculation Examples
Example 1
60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5. Prime factors: two 2s, one 3, one 5.
Example 2
97 is a prime number; its only prime factor is itself.
Frequently Asked Questions
What is the Fundamental Theorem of Arithmetic?
It states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. This is why prime factorization is so fundamental to number theory.
How do you check if a number is prime?
A prime number has exactly two distinct positive divisors: 1 and itself. To check, test divisibility by primes up to sqrt(n). If no divisor is found, n is prime.