What is prime factorization?

Prime factorization expresses a whole number as a product of its prime factors. Every integer greater than 1 has a unique prime factorization (the Fundamental Theorem of Arithmetic). Example: 360 = 2^3 × 3^2 × 5, meaning 2×2×2×3×3×5 = 360. To factor a number, divide by 2 as many times as possible, then by 3, 5, 7, and so on up to the square root.

How do you find the prime factorization of a large number?

Use the trial division method: start dividing by 2 (the smallest prime), recording each factor, then try 3, 5, 7, 11, and so on up to the square root of the remaining value. Any remainder greater than 1 after this process is itself prime. For example: 1260 ÷ 2 = 630 ÷ 2 = 315 ÷ 3 = 105 ÷ 3 = 35 ÷ 5 = 7. So 1260 = 2^2 × 3^2 × 5 × 7.

What is the difference between factors and prime factors?

Factors (or divisors) are all whole numbers that divide evenly into a given number. Prime factors are the subset of factors that are prime numbers. For 12: all factors are 1, 2, 3, 4, 6, 12; prime factors are only 2 and 3. Prime factorization (2^2 × 3) is the canonical form that uniquely identifies any integer.

How do you use prime factorization to find GCF?

To find the GCF (Greatest Common Factor) of two numbers, write out both prime factorizations, then multiply together the prime factors they share, using the smaller exponent for each. Example: GCF of 48 and 180. 48 = 2^4 × 3; 180 = 2^2 × 3^2 × 5. Shared primes: 2 (min exp = 2) and 3 (min exp = 1). GCF = 2^2 × 3 = 12.

How do you use prime factorization to find LCM?

To find the LCM (Least Common Multiple) of two numbers, write out both prime factorizations, then multiply together all prime factors that appear in either number, using the larger exponent for each. Example: LCM of 12 and 18. 12 = 2^2 × 3; 18 = 2 × 3^2. All primes: 2 (max exp = 2) and 3 (max exp = 2). LCM = 2^2 × 3^2 = 36.

Is every number uniquely factorable into primes?

Yes — the Fundamental Theorem of Arithmetic states that every integer greater than 1 has exactly one prime factorization (ignoring the order of factors). This is why prime factorization is such a foundational concept in number theory. The only exception is 1, which has no prime factors. Primes themselves are already factored (e.g. 7 = 7^1).

What are practical applications of prime factorization?

Prime factorization is used in: (1) Finding GCF and LCM — essential for fraction simplification and algebra. (2) Cryptography — RSA encryption relies on the difficulty of factoring very large numbers into primes. (3) Simplifying fractions — divide numerator and denominator by their GCF. (4) Solving problems in competitive math, number theory, and modular arithmetic.

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Prime Factorization Calculator

Prime factorization expresses a number as a product of prime numbers. Every integer > 1 has a unique prime factorization (Fundamental Theorem of Arithmetic). Example: 360 = 2^3 x 3^2 x 5. The trial division algorithm divides by each prime in sequence. Use prime factorization to find GCF (Greatest Common Factor) by taking minimum exponents, and LCM (Least Common Multiple) by taking maximum exponents of shared primes.

Find the prime factorization of any number up to 10,000,000. Shows prime factors with exponents, all divisors, and checks if the number is prime. Also calculates GCF (Greatest Common Factor) and LCM (Least Common Multiple) for two numbers.

Mode

Quick Examples

Enter a Number

Enter any integer from 1 to 10,000,000

How to Use

1
Enter a number
Type any integer from 1 to 10,000,000 in the input field. Or click one of the Quick Examples (12, 360, 1024, 997, 9999) to see a demonstration.
2
View the factorization
The result shows the prime factorization in canonical form with exponents (e.g. 360 = 2^3 x 3^2 x 5) and the full expansion (2 x 2 x 2 x 3 x 3 x 5). If the number is prime, a green badge indicates it.
3
Check all divisors
Scroll down to see every whole number that divides evenly into the result. The count shows how many divisors the number has (perfect squares have an odd number of divisors).
4
Calculate GCF and LCM
Switch to GCF & LCM mode to enter two numbers. The calculator shows the factorization of both and computes the Greatest Common Factor and Least Common Multiple simultaneously.

Frequently Asked Questions

What is prime factorization?: Prime factorization expresses a whole number as a product of its prime factors. Every integer greater than 1 has a unique prime factorization (the Fundamental Theorem of Arithmetic). Example: 360 = 2^3 × 3^2 × 5, meaning 2×2×2×3×3×5 = 360. To factor a number, divide by 2 as many times as possible, then by 3, 5, 7, and so on up to the square root.
How do you find the prime factorization of a large number?: Use the trial division method: start dividing by 2 (the smallest prime), recording each factor, then try 3, 5, 7, 11, and so on up to the square root of the remaining value. Any remainder greater than 1 after this process is itself prime. For example: 1260 ÷ 2 = 630 ÷ 2 = 315 ÷ 3 = 105 ÷ 3 = 35 ÷ 5 = 7. So 1260 = 2^2 × 3^2 × 5 × 7.
What is the difference between factors and prime factors?: Factors (or divisors) are all whole numbers that divide evenly into a given number. Prime factors are the subset of factors that are prime numbers. For 12: all factors are 1, 2, 3, 4, 6, 12; prime factors are only 2 and 3. Prime factorization (2^2 × 3) is the canonical form that uniquely identifies any integer.
How do you use prime factorization to find GCF?: To find the GCF (Greatest Common Factor) of two numbers, write out both prime factorizations, then multiply together the prime factors they share, using the smaller exponent for each. Example: GCF of 48 and 180. 48 = 2^4 × 3; 180 = 2^2 × 3^2 × 5. Shared primes: 2 (min exp = 2) and 3 (min exp = 1). GCF = 2^2 × 3 = 12.
How do you use prime factorization to find LCM?: To find the LCM (Least Common Multiple) of two numbers, write out both prime factorizations, then multiply together all prime factors that appear in either number, using the larger exponent for each. Example: LCM of 12 and 18. 12 = 2^2 × 3; 18 = 2 × 3^2. All primes: 2 (max exp = 2) and 3 (max exp = 2). LCM = 2^2 × 3^2 = 36.
Is every number uniquely factorable into primes?: Yes — the Fundamental Theorem of Arithmetic states that every integer greater than 1 has exactly one prime factorization (ignoring the order of factors). This is why prime factorization is such a foundational concept in number theory. The only exception is 1, which has no prime factors. Primes themselves are already factored (e.g. 7 = 7^1).
What are practical applications of prime factorization?: Prime factorization is used in: (1) Finding GCF and LCM — essential for fraction simplification and algebra. (2) Cryptography — RSA encryption relies on the difficulty of factoring very large numbers into primes. (3) Simplifying fractions — divide numerator and denominator by their GCF. (4) Solving problems in competitive math, number theory, and modular arithmetic.