Prime Factorization Calculator

Prime factorization decomposes an integer into the product of its prime factors. This calculator finds all prime factors of a number using successive prime division.

Prime factorization method

The algorithm tests prime divisors in ascending order:

• Divide the number by 2 as many times as possible.
• Then try 3, 5, 7, 11, etc., until the quotient is 1.
• You only need to test primes up to the square root of the number.

The result is expressed as a product of prime powers, for example 60 = 2² × 3 × 5.

Example 1: factorizing 60

Problem: Factorize 60 into prime factors.

1. Divide by 2:
   • 60 ÷ 2 = 30; 30 ÷ 2 = 15. (two factors of 2)
2. Divide by 3:
   • 15 ÷ 3 = 5. (one factor of 3)
3. Divide by 5:
   • 5 ÷ 5 = 1. (one factor of 5)

Answer: 60 = 2² × 3 × 5.

Example 2: factorizing 315

Problem: Factorize 315 into prime factors.

1. Divide by 3:
   • 315 ÷ 3 = 105; 105 ÷ 3 = 35. (two factors of 3)
2. Divide by 5:
   • 35 ÷ 5 = 7. (one factor of 5)
3. Divide by 7:
   • 7 ÷ 7 = 1. (one factor of 7)

Answer: 315 = 3² × 5 × 7.

Common uses of prime factorization

• Computing GCF and LCM of large numbers from their prime factors.
• Simplifying fractions and radicals in algebra.
• Understanding the fundamental structure of numbers in number theory.
• Foundation of RSA cryptography and security systems.
• Solving divisibility and congruence problems.
• Verifying whether a number is prime or composite.

Common mistakes in prime factorization

• Including composite numbers as factors (e.g., 4 instead of 2²).
• Not verifying that all factors are actually prime.
• Stopping before reaching 1, leaving factors undecomposed.
• Not testing enough primes, especially for large numbers.

Pro tip

The Fundamental Theorem of Arithmetic guarantees that every integer greater than 1 has a unique prime factorization. This means no matter which method you use, you will always arrive at the same prime factors.