What is prime factorization?

Prime factorization is the process of finding which prime numbers multiply together to give the original number. Every integer greater than 1 has a unique prime factorization.

What is a prime number?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples: 2, 3, 5, 7, 11, 13.

Why is prime factorization useful?

It is fundamental in mathematics — used for finding GCD and LCM, simplifying fractions, cryptography (RSA), and number theory.

How are the total divisors calculated?

If n = p1^a1 x p2^a2 x ... then the number of divisors is (a1+1)(a2+1)... For example, 360 = 2^3 x 3^2 x 5^1 has (3+1)(2+1)(1+1) = 24 divisors.

What is the largest number supported?

This calculator supports numbers up to about 1 trillion (999,999,999,999). Larger numbers may take longer to factor.

MATMathematics

Prime Factorization

Find the prime factors of any number. See the complete prime factorization, check if a number is prime, and count total divisors.

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How It Works

Formula

n = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}

d(n) = (a_1 + 1)(a_2 + 1) \cdots (a_k + 1)

Where

$n$: The integer being factored (n ≥ 2)
$p_i$: The i-th distinct prime factor of n
$a_i$: The exponent of the i-th prime in the factorization
$k$: Number of distinct prime factors
$d(n)$: Total number of positive divisors of n

The calculator uses trial division: it divides the number by each integer starting from 2, counting how many times each prime divides evenly. The process continues until the remaining quotient is 1.

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