Prime Factorization Calculator
Break any number into its prime factors with a step-by-step division table, exponent form, all divisors and a prime-number check.
Enter a whole number from 2 to 999,999,999.
Quick Example
This prime factorization calculator breaks any whole number from 2 to 999,999,999 into its prime factors, showing the result in exponent form, a step-by-step division table and the complete list of divisors. It also tells you whether the number itself is prime, and because it lays out the prime factors of any two numbers it doubles as a quick coprime calculator. Below you will find clear answers to what is a prime number, what is a prime factor, what does coprime mean, plus a handy prime number table.
What Is a Prime Number?
A prime number is a whole number greater than 1 with no positive divisors other than 1 and itself. Prime numbers are the building blocks of every other integer, which is why they are so central to mathematics. The smallest prime number is 2 — also the only even prime — and the first 20 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67 and 71. The number 1 is not prime, because a prime must have exactly two distinct divisors (1 and itself).
Is there a largest prime number? No — Euclid proved around 300 BC that primes are infinite, so they never run out. The largest known prime today is a 41-million-digit Mersenne prime, M136279841, discovered in 2024. There will always be a bigger one waiting to be found.
Prime Number Table (Primes up to 100)
The prime number table below lists every prime up to 100 with the count in each band, a quick reference for homework and divisibility checks.
| Range | Prime Numbers | Count |
|---|---|---|
| 1 – 10 | 2, 3, 5, 7 | 4 |
| 11 – 20 | 11, 13, 17, 19 | 4 |
| 21 – 40 | 23, 29, 31, 37 | 4 |
| 41 – 60 | 41, 43, 47, 53, 59 | 5 |
| 61 – 80 | 61, 67, 71, 73, 79 | 5 |
| 81 – 100 | 83, 89, 97 | 3 |
| Total 1–100 | 25 primes | 25 |
Prime vs Composite Numbers
Every whole number greater than 1 is either prime or composite. A prime number has exactly two divisors, while a composite number has more than two — meaning it can be broken down into smaller prime factors. For instance, 12 is composite (12 = 2² × 3) but 13 is prime. The number 1 belongs to neither group: it has only one divisor. Some primes also come in pairs separated by 2, such as (11, 13) or (17, 19); these are called twin primes and remain one of the famous unsolved problems in number theory.
What Is a Prime Factor?
Prime factors are the prime numbers that divide a number exactly. By the Fundamental Theorem of Arithmetic, every integer greater than 1 can be written as a product of primes in exactly one way — its unique prime factorisation. A few worked examples:
- 360 = 2³ × 3² × 5
- 72 = 2³ × 3²
- 1001 = 7 × 11 × 13
- 100 = 2² × 5²
How to Do Prime Factorization
- Start dividing by the smallest prime, 2.
- When 2 no longer divides evenly, move to the next prime (3, 5, 7...).
- Continue until the quotient is 1.
- Write any repeated factors in exponent form.
Example: 72 ÷ 2 = 36, 36 ÷ 2 = 18, 18 ÷ 2 = 9, 9 ÷ 3 = 3, 3 ÷ 3 = 1 → 72 = 2³ × 3². The calculator above performs exactly these steps and displays them in the division table so you can follow the working.
What Does Coprime Mean?
Two numbers are coprime (also called relatively prime) if their greatest common divisor is 1 — they share no common factor other than 1. So "what does coprime mean" comes down to having no shared prime factor: 8 = 2³ and 9 = 3² are coprime numbers because they have no prime factor in common, even though neither is prime itself.
To use this tool as a coprime numbers checker, factor both numbers and compare their prime factors. If the two lists share nothing, the numbers are coprime. For example, 35 = 5 × 7 and 48 = 2⁴ × 3 share no prime factor, so they are coprime. Coprimality matters in cryptography (the RSA algorithm relies on coprime exponents), fraction simplification and modular arithmetic.
Uses of Prime Factorization
- GCD and LCM: Found directly from the shared and combined prime factors of two numbers.
- Fraction simplification: Cancel common prime factors in the numerator and denominator.
- Cryptography: RSA security relies on the difficulty of factoring very large numbers.
- Number theory: The foundation for divisibility rules, perfect numbers and many proofs.
Enter any number above to see its full prime factorisation, division steps and divisors instantly. For more answers, see the frequently asked questions below.
Frequently Asked Questions About the Prime Factorization Calculator
A prime number is a whole number greater than 1 with no positive divisors other than 1 and itself. The smallest prime is 2 (the only even prime). The first ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. The number 1 is not prime.
Prime factors are the prime numbers that divide a number exactly. By the Fundamental Theorem of Arithmetic, every integer greater than 1 can be written as a product of primes in exactly one way. Example: 360 = 2³ × 3² × 5.
Two numbers are coprime if their GCD is 1 — they share no common factor other than 1. Example: 8 = 2³ and 9 = 3² share no prime factors, so they are coprime.
Steps: 1) Start dividing by 2. 2) Move to the next prime when 2 no longer divides. 3) Continue until the quotient is 1. 4) Write repeated factors in exponent form. Example: 72 = 2³ × 3².
The smallest prime is 2. There is no largest prime — Euclid proved primes are infinite. The largest known prime (2024) is the 41-million-digit Mersenne prime M136279841.
Calculate their GCD; if it equals 1 they are coprime. Or factor both numbers — if they share no prime factor they are coprime. Example: 35 = 5×7, 48 = 2⁴×3 → coprime.
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