Prime Factorization Calculator
Break any number into its prime factors using the Fundamental Theorem of Arithmetic. Includes a visual factor tree showing how the number branches into primes.
Prime Factors
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The Formula
Every integer greater than 1 can be uniquely expressed as a product of prime numbers raised to exponents. This is the Fundamental Theorem of Arithmetic, which guarantees that every number has exactly one prime factorization (ignoring the order of factors). This uniqueness is what makes prime numbers the fundamental building blocks of all integers.
Variable Definitions
Prime Factors
Prime numbers that divide n evenly. Primes are numbers greater than 1 with exactly two factors: 1 and themselves.
Exponents
How many times each prime factor appears in the factorization. For example, in 72 = 2 x 2 x 2 x 3 x 3, the exponent of 2 is 3 and the exponent of 3 is 2.
How to Use This Calculator
- 1
Enter any positive integer greater than 1 in the input field.
- 2
View the prime factorization as a multiplication expression of prime numbers with exponents.
- 3
See the interactive factor tree visualization that shows how the number breaks down step by step into its prime building blocks.
- 4
Check the total number of prime factors (with multiplicity) to understand the number's "factor depth."
A factor tree breaks a number down step by step until all branches end at prime numbers.
Understanding the Concept
Prime factorization breaks a number into its fundamental prime building blocks. The Fundamental Theorem of Arithmetic states that every integer greater than 1 has a unique prime factorization — meaning there is only one way to write it as a product of primes (ignoring the order). This uniqueness is what makes prime numbers so important: they are the atoms of the number system. To find the prime factorization, divide the number by the smallest prime (2) as many times as possible, then move to the next prime (3), and continue until the result is 1. For example, 72 : divide by 2 three times to get 9, then divide by 3 twice to get 1, giving 72 = 2 x 2 x 2 x 3 x 3 = 2 x 3. The factor tree visualization shows this process as a branching diagram. Prime factorization is essential for finding GCD and LCM (using shared and highest exponents), simplifying radicals, testing whether numbers are perfect squares or cubes, and understanding the structure of the number system. Cryptography, particularly RSA encryption, relies on the fact that factoring large numbers into primes is computationally difficult.
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