Prime Factorization & Factor Tree Calculator
Find prime factors, exponential prime decomposition, all positive divisors, and factor count.
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Prime Factorization & Factor Tree Guide: Sieve of Eratosthenes
Decomposing Integer 60 into Prime Building Blocks
Fundamental Theorem of Arithmetic: Every integer greater than 1 is either a prime itself or can be uniquely represented as the product of prime numbers.
Prime factorization decomposes a composite integer into a product of prime numbers (numbers greater than 1 that have no positive divisors other than 1 and themselves). Prime numbers serve as the fundamental "atoms" of all number theory and modern cryptography (such as RSA encryption).
1. Canonical Exponential Form
Any positive integer $n$ can be uniquely expressed in canonical prime factorization form:
For example, $360 = 2^3 \times 3^2 \times 5^1$.
2. Divisor Count & Sum Functions
From the prime factorization, you can directly determine the total count of positive divisors $\sigma_0(n)$:
For $360 = 2^3 \times 3^2 \times 5^1$, total divisors = $(3+1)(2+1)(1+1) = 4 \times 3 \times 2 = \mathbf{24\text{ divisors}}$.
🔗 Number Theory & Prime Decompositions
The Fundamental Theorem of Arithmetic establishes that every integer has a unique prime factorization. Browse our ➗ Math & Statistics Hub for number theory tools.
Find Greatest Common Factors or calculate exponential powers with the tools below.
Frequently Asked Questions (Prime Factors)
By mathematical definition, 1 is excluded from the primes to preserve the uniqueness of the Fundamental Theorem of Arithmetic. If 1 were prime, factorizations would not be unique (e.g. $6 = 2 \times 3 = 1 \times 2 \times 3 = 1^5 \times 2 \times 3$).
RSA encryption pairs public keys made from multiplying two massive 1024-bit prime numbers. While multiplying two primes takes milliseconds, finding the prime factors of the resulting product takes supercomputers billions of years.