Prime Factorization Calculator

What Is Prime Factorization?

Prime factorization is the process of writing a whole number greater than 1 as a product of prime numbers. A prime number has exactly two positive factors: 1 and itself. Every integer greater than 1 has one unique prime factorization, apart from the order in which the factors are written.

Prime factorization is useful for working with factors, divisibility, greatest common factors (GCF), least common multiples (LCM), and other number theory problems. A factor tree is one common way to break a number down until every factor is prime.

How to Find Prime Factors

1. 1. Start: Start with the number you want to factor.
2. 2. Divide: Divide it by the smallest prime number that divides it evenly, usually 2.
3. 3. Continue: Continue dividing the resulting quotient by prime numbers.
4. 4. Stop: Stop when the remaining quotient is prime.
5. 5. Verify: Multiply all the prime factors together to verify that they produce the original number.
6. 6. Use Exponents: If a prime factor occurs more than once, it can be written using an exponent.

Example: Prime Factorization of 60

Step 1: Start with 60. The smallest prime number that divides 60 is 2.

Step 2: The quotient is 30, which is also divisible by 2.

Step 3: The quotient is now 15. The next prime number that divides 15 is 3.

Step 4: The remaining number, 5, is prime, so the factorization is complete.

Since 2 appears twice, the repeated factor can be written with an exponent.

Using a Factor Tree

A factor tree starts by splitting a composite number into two factors. Each composite factor is split again until all the branches end in prime numbers.

For 60, one possible factor tree begins with 6 and 10. Both numbers can then be split into prime factors:

Reading the prime numbers at the ends gives the same prime factorization:

Prime Factorization With Exponents

When the same prime factor appears several times, exponents provide a shorter way to write the result. The exponent tells you how many times that prime factor is multiplied by itself.

For example, the prime factorization of 360 contains three factors of 2, two factors of 3, and one factor of 5:

How to Check a Prime Factorization

You can verify a prime factorization by multiplying all of its prime factors together. For example, check the factorization of 84:

Expand the exponent and multiply:

Because the product is 84, the factorization is correct. Each factor is also prime, so the prime factorization is complete.

Special Cases

Prime numbers: If the starting number is already prime, its prime factorization is the number itself. For example, 13 cannot be broken into smaller positive factors other than 1 and 13, so its prime factorization is simply 13.

Number 1: The number 1 is neither prime nor composite, so it does not have a prime factorization.

Zero: Prime factorization is normally defined for positive integers greater than 1. Zero is not assigned a prime factorization.

Common Mistakes to Avoid

• Stopping at a composite factor: Continue factoring until every factor is prime.
• Including 1 as a prime factor: 1 is not a prime number and should not appear in a prime factorization.
• Forgetting repeated factors: If a prime divides the number more than once, include it each time or use an appropriate exponent.
• Using a non-prime factor as the final factor: A complete prime factorization must contain only prime numbers.
• Not checking the result: Multiply the prime factors together to confirm that they equal the original number.
• Confusing factors with prime factors: A number can have composite factors, but prime factorization specifically breaks the number down into prime factors.

Why Use a Prime Factorization Calculator?

A Prime Factorization Calculator can quickly break an integer into its prime factors and show the factorization step by step. It can be useful for checking manual work, working with larger numbers, and seeing repeated prime factors in exponential form.