How do I use the Prime Factorization Calculator?

Enter the required values, then calculate. The Prime Factorization Calculator shows results, formulas, and step-by-step work when available. No account is required.

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How it works

Prime factorization writes n as a product of primes. Methods include successive trial division and factor trees; the multiset of primes is unique for n > 1.

n = p₁ × p₂ × … = p₁^e₁ × p₂^e₂ × …

When to use

Use this tool when you need to: try 820 for the calculator.net walkthrough, or 60 for 2 × 2 × 3 × 5. Related: Factor and Common Factor.

How to use this tool

Enter your values in the fields above.
Review the formula and any mode options for your problem.
Read the result and the step-by-step solution.

Try 820 for the calculator.net walkthrough, or 60 for 2 × 2 × 3 × 5. Related: Factor and Common Factor.

Frequently asked questions

What is the Prime Factorization Calculator?: Prime factorization writes n as a product of primes. Methods include successive trial division and factor trees; the multiset of primes is unique for n > 1.
How do I use the Prime Factorization Calculator?: Enter the required values, then calculate. The Prime Factorization Calculator shows results, formulas, and step-by-step work when available. No account is required.
What formula does the Prime Factorization Calculator use?: n = p₁ × p₂ × … = p₁^e₁ × p₂^e₂ × …

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Prime factorization

Enter a non-zero integer to get its prime factors (expanded and compact), trial-division steps, and a factor tree.

How it works

What is a prime number?

Prime numbers are natural numbers greater than 1 that cannot be formed by multiplying two smaller natural numbers greater than 1. For example, 7 can only be written as 1 × 7 among positive integers — so 7 is prime. Other primes include 2, 3, 5, 11, and so on. Natural numbers greater than 1 that are not prime are composite (for example 4, 6, 9). The number 1 is neither prime nor composite.

What is prime factorization?

Prime factorization decomposes a composite number into a product of primes. By the fundamental theorem of arithmetic, every natural number greater than 1 is either prime or can be written uniquely (up to order) as a product of primes. Example: 60 = 5 × 3 × 2 × 2 (no composites remain in the product). That is the same as the compact form 2² × 3 × 5.

Trial division

Trial division tests successive primes: divide out 2 as many times as possible, then 3, 5, and so on, until the remaining value is prime. Example — factor 820: 820 ÷ 2 = 410 410 ÷ 2 = 205 205 is not divisible by 2 or 3, but 205 ÷ 5 = 41 41 is prime, so stop. 820 = 2 × 2 × 5 × 41 = 2² × 5 × 41. This “brute force” method is simple for modest numbers and becomes tedious for very large composites.

Prime decomposition and factor trees

A factor tree breaks a composite into a factor pair and continues on any composite branch until every leaf is prime. Different trees can rearrange the order of splits, but the multiset of prime leaves is the same. Example paths for 820 both finish with primes 2, 2, 5, and 41: 820 = 2 × 410; 410 = 2 × 205; 205 = 5 × 41 Very large integers can take a long time even for computers — for example, factoring RSA-768 (232 digits) took years of distributed work — so there is no practical all-purpose method that is both simple and fast for every enormous number.

Prime factorization of common numbers

Quick reference (compact form): 2–3, 5, 7, 11, … — prime 4 = 2² · 6 = 2 × 3 · 8 = 2³ · 9 = 3² · 10 = 2 × 5 12 = 2² × 3 · 16 = 2⁴ · 18 = 2 × 3² · 20 = 2² × 5 24 = 2³ × 3 · 30 = 2 × 3 × 5 · 36 = 2² × 3² · 48 = 2⁴ × 3 60 = 2² × 3 × 5 · 100 = 2² × 5² · 120 = 2³ × 3 × 5 200 = 2³ × 5² · 500 = 2² × 5³ · 1000 = 2³ × 5³ Enter any integer in the calculator above for a full workup.

Related calculators

Factor lists every positive divisor of one number; common factors find shared divisors across a list; GCF and LCM combine factors of several numbers.

How it works

Prime factorization writes n as a product of primes. Methods include successive trial division and factor trees; the multiset of primes is unique for n > 1.

n = p₁ × p₂ × … = p₁^e₁ × p₂^e₂ × …

When to use

Use this tool when you need to: try 820 for the calculator.net walkthrough, or 60 for 2 × 2 × 3 × 5. Related: Factor and Common Factor.

How to use this tool

Enter your values in the fields above.
Review the formula and any mode options for your problem.
Read the result and the step-by-step solution.

Try 820 for the calculator.net walkthrough, or 60 for 2 × 2 × 3 × 5. Related: Factor and Common Factor.

Frequently asked questions

What is the Prime Factorization Calculator?: Prime factorization writes n as a product of primes. Methods include successive trial division and factor trees; the multiset of primes is unique for n > 1.
How do I use the Prime Factorization Calculator?: Enter the required values, then calculate. The Prime Factorization Calculator shows results, formulas, and step-by-step work when available. No account is required.
What formula does the Prime Factorization Calculator use?: n = p₁ × p₂ × … = p₁^e₁ × p₂^e₂ × …