Prime Factorization Calculator (2024)

Created by Mateusz Mucha, Piotr Małek and Hanna Pamuła, PhD

Reviewed by

Bogna Szyk and Jack Bowater

Last updated:

Nov 20, 2023

Table of contents:
  • What is a prime number?
  • What is a prime factor?
  • What is prime factorization?
  • How to find prime factorization? A factor tree method
  • Greatest Common Factor
  • Least Common Multiple
  • What is prime factorization of...

The prime factorization calculator will take any number and find its prime factors. Simply type the number into our tool and in no time you'll find the prime factorization. To understand the whole process, first you must get familiar with what is a prime factor. Once you understand that, we will move on to the difference between prime factor and prime factorization.

Below, you'll find all the answers, as well as concise information about how to find prime factorization and what a factor tree is.

What is a prime number?

To understand prime factorization, we need to start from the beginning - what is a prime number? A prime numbers are numbers whose only factors are one and itself - in other words, it can't be formed by multiplying two smaller natural numbers. A key point to note is that the two factors must be different, so 1 is not a prime number since both factors of 1 are the same. For example, 5 is a prime number since the only factors of 5 are 1 and 5. 6 is not a prime as, apart from 1 and 6, other factors exist - 2 and 3.

There are infinitely many primes and there's no simple formula to determine whether a number is a prime or not. That's why this prime factorization calculator is such a great, multi-purpose tool - you can use it as a prime number calculator as well!

What is a prime factor?

Prime factors are factors of a number that are themselves prime numbers. For example, suppose we want to find the factors of 20, that is, we want to know what whole numbers multiply to give you 20. We know that 1 * 20 = 20, 2 * 10 = 20 and 4 * 5 = 20. But notice that 20, 10 and 4 are not prime factors. The only prime factors of 20 are 2 and 5.

What is prime factorization?

Prime factorization is when we break a number down into factors that are only prime numbers. If we look at the above example with 20, the factors are 1, 2, 4, 5, 10, 20. The best place to start is to find at least one initial factor that is prime. Since 5 is prime, we can start with 4 * 5. Notice that 4 is not prime, so we break 4 down into 2 * 2. Since 2 is prime, the prime factorization of 20 is 2 * 2 * 5. Go ahead and check this result with our prime factorization calculator.

How to find prime factorization? A factor tree method

We'll show you step by step how to find prime factorization. We'll use the factor tree diagram, an easy way to break down a number into its prime factors. Are you ready?

  1. Take a number. There's no point in picking a prime number, as the prime factorization will finish at this point. Let's choose 36.
  2. Factor it into any two numbers, prime or not. You may want to take the easiest splits, e.g., if your number is even, split it into 2 and the other number. 36 is even so that we can write it as 2 * 18.
  3. Start constructing the factor tree. Draw two branches splitting down from your original number.
Prime Factorization Calculator (1)
  1. Factor the next line. If your number is a prime, leave it as it is. If it's not a prime number, repeat step 2.
Prime Factorization Calculator (2)
  1. Repeat step 4 until you're left with only prime numbers.
Prime Factorization Calculator (3)
  1. Write down the final prime factorization and prime factors.
Prime Factorization Calculator (4)

Take all the "leaves" of your factor tree and multiply them together:

36 = 2 * 2 * 3 * 3

That's how we find prime factorization!

The prime factors of our original number 36 are 2 and 3. The same prime factor may occur more than once, which is precisely what happened in our case - both prime numbers appear twice in the prime factorization. We can then express it as:

36 = 2² * 3²

Of course, you should get the same result with a different factor tree splits, as e.g. here:

Prime Factorization Calculator (5)

Greatest Common Factor

Prime factorization is the first step in finding the greatest common factor - the greatest factor of two or more numbers. The GCF is especially useful for simplifying fractions and solving equations using polynomials. For example, the greatest common factor between 6 and 20 is 2: prime factorization of the first number is 6 = 2 * 3, the latter may be expressed as 20 = 5 * 2 * 2, and the only number which appears in both prime factorizations is 2, indeed. If you want to calculate greatest common factor in a snap, use our greatest common factor calculator.

Least Common Multiple

The the prime factorization calculator is also useful in finding the least common multiple (LCM). The LCM is important when adding fractions with different denominators. The least common multiple is obtained when you multiply the higher powers of all factors between the two numbers. For example, the least common multiple between 6 and 20 is (2 * 2 * 3 * 5) = 60. The LCM may be found by hand or with use of the least common multiple calculator.

What is prime factorization of...

Here is a list of prime factorizations for ease of reference. You can verify these with our prime factorization calculator.

  • 2 is prime

  • 3 is prime

  • 4 = 2 × 2

  • 5 is prime

  • 6 = 2 × 3

  • 7 is prime

  • 8 = 2 × 2 × 2

  • 9 = 3 × 3

  • 10 = 2 × 5

  • 11 is prime

  • 12 = 2 × 2 × 3

  • 13 is prime

  • 14 = 2 × 7

  • 15 = 3 × 5

  • 16 = 2 × 2 × 2 × 2

  • 17 is prime

  • 18 = 2 × 3 × 3

  • 19 is prime

  • 20 = 2 × 2 × 5

  • 21 = 3 × 7

  • 22 = 2 × 11

  • 23 is prime

  • 24 = 2 × 2 × 2 × 3

  • 25 = 5 × 5

  • 26 = 2 × 13

  • 27 = 3 × 3 × 3

  • 28 = 2 × 2 × 7

  • 29 is prime

  • 30 = 2 × 3 × 5

  • 31 is prime

  • 32 = 2 × 2 × 2 × 2 × 2

  • 33 = 3 × 11

  • 34 = 2 × 17

  • 35 = 5 × 7

  • 36 = 2 × 2 × 3 × 3

  • 37 is prime

  • 38 = 2 × 19

  • 39 = 3 × 13

  • 40 = 2 × 2 × 2 × 5

  • 41 is prime

  • 42 = 2 × 3 × 7

  • 43 is prime

  • 44 = 2 × 2 × 11

  • 45 = 3 × 3 × 5

  • 46 = 2 × 23

  • 47 is prime

  • 48 = 2 × 2 × 2 × 2 × 3

  • 49 = 7 × 7

  • 50 = 2 × 5 × 5

  • 51 = 3 × 17

  • 52 = 2 × 2 × 13

  • 53 is prime

  • 54 = 2 × 3 × 3 × 3

  • 55 = 5 × 11

  • 56 = 2 × 2 × 2 × 7

  • 57 = 3 × 19

  • 58 = 2 × 29

  • 59 is prime

  • 60 = 2 × 2 × 3 × 5

  • 61 is prime

  • 62 = 2 × 31

  • 63 = 3 × 3 × 7

  • 64 = 2 × 2 × 2 × 2 × 2 × 2

  • 65 = 5 × 13

  • 66 = 2 × 3 × 11

  • 67 is prime

  • 68 = 2 × 2 × 17

  • 69 = 3 × 23

  • 70 = 2 × 5 × 7

  • 71 is prime

  • 72 = 2 × 2 × 2 × 3 × 3

  • 73 is prime

  • 74 = 2 × 37

  • 75 = 3 × 5 × 5

  • 76 = 2 × 2 × 19

  • 77 = 7 × 11

  • 78 = 2 × 3 × 13

  • 79 is prime

  • 80 = 2 × 2 × 2 × 2 × 5

  • 81 = 3 × 3 × 3 × 3

  • 82 = 2 × 41

  • 83 is prime

  • 84 = 2 × 2 × 3 × 7

  • 85 = 5 × 17

  • 86 = 2 × 43

  • 87 = 3 × 29

  • 88 = 2 × 2 × 2 × 11

  • 89 is prime

  • 90 = 2 × 3 × 3 × 5

  • 91 = 7 × 13

  • 92 = 2 × 2 × 23

  • 93 = 3 × 31

  • 94 = 2 × 47

  • 95 = 5 × 19

  • 96 = 2 × 2 × 2 × 2 × 2 × 3

  • 97 is prime

  • 98 = 2 × 7 × 7

  • 99 = 3 × 3 × 11

  • 100 = 2 × 2 × 5 × 5

  • 101 is prime

  • 102 = 2 × 3 × 17

  • 103 is prime

  • 104 = 2 × 2 × 2 × 13

  • 105 = 3 × 5 × 7

  • 108 = 2 × 2 × 3 × 3 × 3

  • 117 = 3 × 3 × 13

  • 120 = 2 × 2 × 2 × 3 × 5

  • 121 = 11 × 11

  • 125 = 5 × 5 × 5

  • 126 = 2 × 3 × 3 × 7

  • 130 = 2 × 5 × 13

  • 132 = 2 × 2 × 3 × 11

  • 135 = 3 × 3 × 3 × 5

  • 140 = 2 × 2 × 5 × 7

  • 144 = 2 × 2 × 2 × 2 × 3 × 3

  • 147 = 3 × 7 × 7

  • 150 = 2 × 3 × 5 × 5

  • 162 = 2 × 3 × 3 × 3 × 3

  • 175 = 5 × 5 × 7

  • 180 = 2 × 2 × 3 × 3 × 5

  • 196 = 2 × 2 × 7 × 7

  • 200 = 2 × 2 × 2 × 5 × 5

  • 210 = 2 × 3 × 5 × 7

  • 216 = 2 × 2 × 2 × 3 × 3 × 3

  • 225 = 3 × 3 × 5 × 5

  • 245 = 5 × 7 × 7

  • 250 = 2 × 5 × 5 × 5

  • 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

  • 300 = 2 × 2 × 3 × 5 × 5

  • 375 = 3 × 5 × 5 × 5

  • 400 = 2 × 2 × 2 × 2 × 5 × 5

  • 500 = 2 × 2 × 5 × 5 × 5

  • 625 = 5 × 5 × 5 × 5

At the beginning, 1 was considered a prime number. It was not until the early 20th century that most mathematicians excluded 1 as a prime number. Notice the prime factorization calculator does not include 1 in the results of prime numbers.

Mateusz Mucha, Piotr Małek and Hanna Pamuła, PhD

Prime Factorization Calculator (2024)

FAQs

What is an easy way to check a prime factorization? ›

The simplest algorithm to find the prime factors of a number is to keep on dividing the original number by prime factors until we get the remainder equal to 1. For example, prime factorizing the number 30 we get, 30/2 = 15, 15/3 = 5, 5/5 = 1. Since we received the remainder, it cannot be further factorized.

Why is prime factorization hard? ›

Hence to find the first factor our algorithm has to check about 10 times more numbers. As our product is bigger and the numbers we use to check are bigger, each check takes more time on average. So, we see that adding a few digits on to our prime numbers makes factoring the product much, much harder.

Is there an algorithm for prime factorization? ›

A prime factorization algorithm typically involves testing whether each factor is prime each time a factor is found. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist.

What type of math is prime factorization? ›

Prime factorization is a process of writing all numbers as a product of primes. So, for example, say if we have something like the number 20. We can break that down into two factors. We can say, “well, that's 4 times 5.” And notice, 5 is a prime number.

What are the rules of prime factorization? ›

Prime Factorization by Division Method
  • Step 1: Divide the number by the smallest prime number such that the smallest prime number should divide the number completely. ...
  • Step 2: Again, divide the quotient of step 1 by the smallest prime number. ...
  • Step 3: Repeat step 2, until the quotient becomes 1.

Why is 1 not a prime number? ›

The answer to this lies in the definition of prime numbers itself. For a number to be called as a prime number, it must have only two positive factors. Now, for 1, the number of positive divisors or factors is only one i.e. 1 itself. So, number one is not a prime number.

How do you answer factorization? ›

List of Factorization Formulas for Algebraic Equation
  1. (a + b)2 = a2 + 2ab + b2
  2. (a − b)2 = a2 − 2ab + b2
  3. (a + b)3 =a3 + b3 + 3ab(a + b)
  4. (a – b)3 = a3 – b3 – 3ab(a – b)
  5. (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
  6. (a − b)4 = a4 − 4a3b + 6a2b2 − 4ab3 + b4
  7. (a + b + c)2 = a2 + b2 +c2 + 2ab + 2ac + 2bc.
  8. a2 – b2 = (a + b)(a – b)

What is the trick to find prime numbers? ›

To find whether a larger number is prime or not, add all the digits in a number, if the sum is divisible by 3 it is not a prime number. Except 2 and 3, all the other prime numbers can be expressed in the general form as 6n + 1 or 6n - 1, where n is the natural number.

What are the three methods of prime factorization? ›

Step 1: First divide the given number by the smallest prime number. In this case, the smallest prime number should be taken in such a way that it divides the number exactly. Step 2: Again, divide the quotient by the possible smallest prime number. Step 3: Repeat the same process, until the quotient becomes 1.

How to simplify prime factorization? ›

To use prime factorization to reduce a fraction, we follow these steps.
  1. Replace the numerator and denominator with their prime factorizations.
  2. Cancel out any common factors in the numerator and denominator.
  3. Multiply any leftover factors in the numerator together and any leftover factors in the denominator together.

How do you solve factorization easily? ›

The simplest way of factorising is:
  1. Find the highest common factor of each of the terms in the expression.
  2. Write the highest common factor (HCF) in front of any brackets.
  3. Fill in each term in the brackets by multiplying out.

What is the fastest way to find the factors of a number? ›

The fastest way to find the factors of a number is to divide the number by the smallest prime number (bigger than 1). This needs to go into it evenly without any remainder. Go on with this process with each number you get, until you reach 1.

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