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Prime Factorization Calculator

Decompose any positive integer into its canonical prime powers, explore trial division steps, inspect all positive divisors, and verify primality per the Fundamental Theorem of Arithmetic.

Enter an integer greater than 1 (e.g. 720, 2048, 1000000)
PRIME FACTORIZATION (EXPONENT FORM)
2⁴ × 3² × 5
Unique prime product
CLASSIFICATION
Composite Number
Distinct primes: 2, 3, 5
TOTAL DIVISORS (τ)
30
(4+1)(2+1)(1+1) = 30
SUM OF DIVISORS (σ)
2,418
Sum of all positive factors
ALL POSITIVE DIVISORS OF 720
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720

Trial Division Factorization Steps

The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic, established in Euclid's Elements and proved formally by Carl Friedrich Gauss in Disquisitiones Arithmeticae (1801), states that every integer $n > 1$ can be uniquely factored into a product of prime powers:

n = p₁^{a₁} · p₂^{a₂} · p₃^{a₃} ... p_k^{a_k}

Calculating the Number of Divisors τ(n)

Any divisor of $n$ must be composed of the same prime factors with exponents ranging from $0$ up to $a_i$. Thus, the total count of distinct positive divisors is:

τ(n) = (a₁ + 1)(a₂ + 1)(a₃ + 1)...(a_k + 1)

Frequently Asked Questions

What is trial division?
Trial division is the simplest primality testing and integer factorization algorithm. It tests whether an integer n can be divided evenly by each prime number up to the square root of n (sqrt(n)). If no factor is found up to sqrt(n), the remaining quotient is guaranteed to be prime.
What is a prime factor tree?
A factor tree is a graphical breakdown where a composite number branches into two factors. The branches continue dividing until all leaf nodes are prime numbers. Multiplying all leaf nodes together reconstructs the original number.