Prime Factorization Calculator

What Is Prime Factorization?

Prime factorization is the process of expressing an integer greater than 1 as a product of prime numbers. Every integer greater than 1 has a unique prime factorization, apart from the order in which the factors are written.

For example, 60 can be broken down into prime factors as follows:

Using exponent notation, the same factorization can be written as:

How to Find the Prime Factors of a Number

A common method for finding prime factors is repeated division. Start with the smallest prime number, 2, and divide the integer whenever it divides evenly.

Once 2 is no longer a factor, test the next possible prime factors, such as 3, 5, 7, 11, and so on.

Example: Prime Factorization of 360

Therefore:

Or, using exponent notation:

Prime Numbers and Prime Factors

A prime number is an integer greater than 1 that has exactly two positive divisors: 1 and itself.

Some small prime numbers are:

Prime numbers cannot be factored into smaller positive integers other than 1 and the number itself. For example, 29 is prime, so its prime factorization is simply:

By contrast, 30 is composite because it can be written as a product of smaller integers:

Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be represented as a product of prime numbers in exactly one way, except for the order of the factors.

For example:

Rearranging the factors does not produce a different prime factorization:

Both expressions contain exactly the same prime factors with the same multiplicities.

What Is Prime Factorization Used For?

Prime factorization is an important concept in elementary number theory and has applications throughout mathematics.

• Finding the greatest common divisor (GCD).
• Finding the least common multiple (LCM).
• Simplifying fractions.
• Studying divisibility.
• Determining the number of divisors of an integer.
• Calculating the sum of divisors.
• Working with modular arithmetic and number theory.
• Understanding the structure of composite numbers.

For example, the prime factors of an integer can be used to calculate its number of positive divisors and its sum of divisors.

Prime Factorization and GCD

Prime factorization provides another way to find the greatest common divisor of two numbers.

Consider:

To find the GCD, take the common prime factors using the smallest exponent for each factor:

You can calculate the GCD directly with the GCD Calculator .

Prime Factorization and LCM

Prime factors can also be used to calculate the least common multiple.

When finding an LCM from prime factorizations, use the largest exponent of every prime appearing in the numbers.

Try the LCM Calculator for direct calculations.