Prime Factorization Calculator (2023)

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Please provide an integer to find its prime factors as well as a factor tree.

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What is a prime number?

Prime numbers are natural numbers (positive whole numbers that sometimes include 0 in certain definitions) that are greater than 1, that cannot be formed by multiplying two smaller numbers. An example of a prime number is 7, since it can only be formed by multiplying the numbers 1 and 7. Other examples include 2, 3, 5, 11, etc.

Numbers that can be formed with two other natural numbers, that are greater than 1, are called composite numbers. Examples of this include numbers like, 4, 6, 9, etc.

Prime numbers are widely used in number theory due to the fundamental theorem of arithmetic. This theorem states that natural numbers greater than 1 are either prime, or can be factored as a product of prime numbers. As an example, the number 60 can be factored into a product of prime numbers as follows:

60 = 5 × 3 × 2 × 2

As can be seen from the example above, there are no composite numbers in the factorization.

What is prime factorization?

Prime factorization is the decomposition of a composite number into a product of prime numbers. There are many factoring algorithms, some more complicated than others.

Trial division:

One method for finding the prime factors of a composite number is trial division. Trial division is one of the more basic algorithms, though it is highly tedious. It involves testing each integer by dividing the composite number in question by the integer, and determining if, and how many times, the integer can divide the number evenly. As a simple example, below is the prime factorization of 820 using trial division:

820 ÷ 2 = 410

410 ÷ 2 = 205

Since 205 is no longer divisible by 2, test the next integers. 205 cannot be evenly divided by 3. 4 is not a prime number. It can however be divided by 5:

205 ÷ 5 = 41

Since 41 is a prime number, this concludes the trial division. Thus:

820 = 41 × 5 × 2 × 2

The products can also be written as:

820 = 41 × 5 × 22

This is essentially the "brute force" method for determining the prime factors of a number, and though 820 is a simple example, it can get far more tedious very quickly.

Prime decomposition:

Another common way to conduct prime factorization is referred to as prime decomposition, and can involve the use of a factor tree. Creating a factor tree involves breaking up the composite number into factors of the composite number, until all of the numbers are prime. In the example below, the prime factors are found by dividing 820 by a prime factor, 2, then continuing to divide the result until all factors are prime. The example below demonstrates two ways that a factor tree can be created using the number 820:

Thus, it can be seen that the prime factorization of 820, in either case, again is:

820 = 41 × 5 × 2 × 2

While these methods work for smaller numbers (and there are many other algorithms), there is no known algorithm for much larger numbers, and it can take a long period of time for even machines to compute the prime factorizations of larger numbers; in 2009, scientists concluded a project using hundreds of machines to factor the 232-digit number, RSA-768, and it took two years.

Prime factorization of common numbers

The following are the prime factorizations of some common numbers.

Prime factorization of 2: prime number

Prime factorization of 3: prime number

Prime factorization of 4: 22

Prime factorization of 5: prime number

Prime factorization of 6: 2 × 3

Prime factorization of 7: prime number

Prime factorization of 8: 23

Prime factorization of 9: 32

Prime factorization of 10: 2 × 5

Prime factorization of 11: prime number

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Prime factorization of 12: 22 × 3

Prime factorization of 13: prime number

Prime factorization of 14: 2 × 7

Prime factorization of 15: 3 × 5

Prime factorization of 16: 24

Prime factorization of 17: prime number

Prime factorization of 18: 2 × 32

Prime factorization of 19: prime number

Prime factorization of 20: 22 × 5

Prime factorization of 21: 3 × 7

Prime factorization of 22: 2 × 11

Prime factorization of 23: prime number

Prime factorization of 24: 23 × 3

Prime factorization of 25: 52

Prime factorization of 26: 2 × 13

Prime factorization of 27: 33

Prime factorization of 28: 22 × 7

Prime factorization of 29: prime number

Prime factorization of 30: 2 × 3 × 5

Prime factorization of 31: prime number

Prime factorization of 32: 25

Prime factorization of 33: 3 × 11

Prime factorization of 34: 2 × 17

Prime factorization of 35: 5 × 7

Prime factorization of 36: 22 × 32

Prime factorization of 37: prime number

Prime factorization of 38: 2 × 19

Prime factorization of 39: 3 × 13

Prime factorization of 40: 23 × 5

Prime factorization of 41: prime number

Prime factorization of 42: 2 × 3 × 7

Prime factorization of 43: prime number

Prime factorization of 44: 22 × 11

Prime factorization of 45: 32 × 5

Prime factorization of 46: 2 × 23

Prime factorization of 47: prime number

Prime factorization of 48: 24 × 3

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Prime factorization of 49: 72

Prime factorization of 50: 2 × 52

Prime factorization of 51: 3 × 17

Prime factorization of 52: 22 × 13

Prime factorization of 53: prime number

Prime factorization of 54: 2 × 33

Prime factorization of 55: 5 × 11

Prime factorization of 56: 23 × 7

Prime factorization of 57: 3 × 19

Prime factorization of 58: 2 × 29

Prime factorization of 59: prime number

Prime factorization of 60: 22 × 3 × 5

Prime factorization of 61: prime number

Prime factorization of 62: 2 × 31

Prime factorization of 63: 32 × 7

Prime factorization of 64: 26

Prime factorization of 65: 5 × 13

Prime factorization of 66: 2 × 3 × 11

Prime factorization of 67: prime number

Prime factorization of 68: 22 × 17

Prime factorization of 69: 3 × 23

Prime factorization of 70: 2 × 5 × 7

Prime factorization of 71: prime number

Prime factorization of 72: 23 × 32

Prime factorization of 73: prime number

Prime factorization of 74: 2 × 37

Prime factorization of 75: 3 × 52

Prime factorization of 76: 22 × 19

Prime factorization of 77: 7 × 11

Prime factorization of 78: 2 × 3 × 13

Prime factorization of 79: prime number

Prime factorization of 80: 24 × 5

Prime factorization of 81: 34

Prime factorization of 82: 2 × 41

Prime factorization of 83: prime number

Prime factorization of 84: 22 × 3 × 7

Prime factorization of 85: 5 × 17

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Prime factorization of 86: 2 × 43

Prime factorization of 87: 3 × 29

Prime factorization of 88: 23 × 11

Prime factorization of 89: prime number

Prime factorization of 90: 2 × 32 × 5

Prime factorization of 91: 7 × 13

Prime factorization of 92: 22 × 23

Prime factorization of 93: 3 × 31

Prime factorization of 94: 2 × 47

Prime factorization of 95: 5 × 19

Prime factorization of 96: 25 × 3

Prime factorization of 97: prime number

Prime factorization of 98: 2 × 72

Prime factorization of 99: 32 × 11

Prime factorization of 100: 22 × 52

Prime factorization of 101: prime number

Prime factorization of 102: 2 × 3 × 17

Prime factorization of 103: prime number

Prime factorization of 104: 23 × 13

Prime factorization of 105: 3 × 5 × 7

Prime factorization of 106: 2 × 53

Prime factorization of 107: prime number

Prime factorization of 108: 22 × 33

Prime factorization of 109: prime number

Prime factorization of 110: 2 × 5 × 11

Prime factorization of 111: 3 × 37

Prime factorization of 112: 24 × 7

Prime factorization of 113: prime number

Prime factorization of 114: 2 × 3 × 19

Prime factorization of 115: 5 × 23

Prime factorization of 116: 22 × 29

Prime factorization of 117: 32 × 13

Prime factorization of 118: 2 × 59

Prime factorization of 119: 7 × 17

Prime factorization of 120: 23 × 3 × 5

Prime factorization of 121: 112

Prime factorization of 122: 2 × 61

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Prime factorization of 123: 3 × 41

Prime factorization of 124: 22 × 31

Prime factorization of 125: 53

Prime factorization of 126: 2 × 32 × 7

Prime factorization of 127: prime number

Prime factorization of 128: 27

Prime factorization of 129: 3 × 43

Prime factorization of 130: 2 × 5 × 13

Prime factorization of 131: prime number

Prime factorization of 132: 22 × 3 × 11

Prime factorization of 133: 7 × 19

Prime factorization of 134: 2 × 67

Prime factorization of 135: 33 × 5

Prime factorization of 136: 23 × 17

Prime factorization of 137: prime number

Prime factorization of 138: 2 × 3 × 23

Prime factorization of 139: prime number

Prime factorization of 140: 22 × 5 × 7

Prime factorization of 141: 3 × 47

Prime factorization of 142: 2 × 71

Prime factorization of 143: 11 × 13

Prime factorization of 144: 24 × 32

Prime factorization of 145: 5 × 29

Prime factorization of 146: 2 × 73

Prime factorization of 147: 3 × 72

Prime factorization of 148: 22 × 37

Prime factorization of 149: prime number

Prime factorization of 150: 2 × 3 × 52

Prime factorization of 200: 23 × 52

Prime factorization of 300: 22 × 3 × 52

Prime factorization of 400: 24 × 52

Prime factorization of 500: 22 × 53

Prime factorization of 600: 23 × 3 × 52

Prime factorization of 700: 22 × 52 × 7

Prime factorization of 800: 25 × 52

Prime factorization of 900: 22 × 32 × 52

Prime factorization of 1000: 23 × 53

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FAQs

What is the prime factorization of 75? ›

What is the prime factorization of 75? The prime factorization of 75 is 3 × 5 × 5 or 3 x 52.

What is the prime factorization of 44? ›

The prime factorization of 44 is 2 × 2 × 11 or 22 × 11.

What is the prime factorization of 54? ›

Prime factorisation of 54 is 2 × 3 × 3 × 3.

What is an example of a prime factorization in math? ›

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 is the prime factorization of 72? ›

For example, we can write the number 72 as a product of prime factors: 72 = 2 3 ⋅ 3 2 . The expression 2 3 ⋅ 3 2 is said to be the prime factorization of 72.

What's the prime factorization of 27? ›

From the factor tree we can observe that the prime factorization of 27 is 3 × 3 × 3. This shows us that 3 is the only prime factor of 27.

What's the prime factorization of 56? ›

The prime factorization of 56 is 2 × 2 × 2 × 7 or 23 × 7.

What's the prime factorization of 36? ›

We can write 36 as a product of prime factors: 36 = 2² × 3². The expression 2² × 3² is said to be the prime factorization of 36.

What is the prime factorization of 15? ›

So, the prime factorization of 15 is 3 × 5, where 3 and 5 both are prime numbers.

What is prime factorization of 225? ›

From the factor tree, we can see that the prime factorization of 225 is 3 × 3 × 5 × 5 = $$3^2 \times 5^2$$. This means 3 and 5 are the prime factors of 225.

What's the prime factorization of 50? ›

From the factor tree we can see prime factorization of 50 is 2 × 5 × 5 = 2 × 52. This means 2, 5 and 5 are the prime factors of 50.

What's the prime factorization of 77? ›

Prime Factorisation of 77

Therefore, the prime factors of 77 are 7 and 11.

What is the prime factorization of 18? ›

The prime factorization of 18 is 2 × 3 × 3 or 2 × 32.

What is the prime factorization of 120? ›

What is the prime factorization of 120? The prime factorization of 120 is 2 × 2 × 2 × 3 × 5 or 23 × 3 × 5.

What is the prime factorization of 42? ›

The prime factorization of 42 is 2 × 3 × 7.

How do you simplify using 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.
Nov 23, 2021

What is prime factorization 7th grade? ›

Prime Factorization One way that we can describe a number is by breaking them down into a product of their prime factors. This is called prime factorization. By definition, the prime factorization of a prime number is the number itself, and the prime factorization of 1 is 1.

What is the prime factorization of 55? ›

So the prime factorization of 55 is given by 5 x 11 and the prime factors of 55 are 5 and 11.

What is the prime factorization of 45? ›

So, the prime factorization of 45 is 3 × 3 × 5 or 32 × 5, where 3 and 5 are the prime numbers.

What is the prime factorization of 21? ›

So, the prime factors of 21 are 3 and 7, where 2 and 7 are the prime numbers. Prime factorisation of 21 is 3 × 7.

What's the prime factorization of 48? ›

What is the prime factorization of 48? The prime factorization of 48 is 2×2×2×2×3 or 24 × 3.

What is the prime factorization of 22? ›

What is the prime factorization of 22? The prime factorization of 22 is 2 × 11.

What is prime factorization of 32? ›

What are the prime factors of 32? The prime factorisation of 32 is 2 x 2 x 2 x 2 x 2.

What is the prime factorization of 12? ›

The numbers 2 and 3 are the only prime factors of 12, but a prime factorization of 12 must list the 2 twice — 2 × 2 × 3 (or 22 × 3), because 2 × 3, by itself, doesn't make 12. Though many numbers can be factored in more than one way, their prime factorization is unique!

What is the prime factorization of 28? ›

The prime factors of 28 are 2, 2, and 7.

What is prime factorization of 66? ›

Hence, the number 66 is written as 2 × 3 × 11, and the prime factors of 66 are 2, 3 and 11. Therefore, the prime factorization of 66 is 2 × 3 × 11.

What's the prime factorization of 24? ›

So, the prime factorisation of 24 is 2 × 2 × 2 × 3 or 23 × 3, where 2 and 3 are the prime numbers.

What is the prime factorization of 25? ›

What is the prime factorization of 25? The prime factorization of 25 is 5 × 5 or 52.

What is the prime factorization of 52? ›

What is the prime factorization of 52? The prime factorization of 52 is 2 × 2 × 13 or 22 × 13.

What's the prime factorization of 30? ›

So, the prime factorisation of 30 is 2 × 3 × 5, where 2, 3 and 5 are the prime numbers.

What is the prime factorization of 76? ›

The prime factorisation of 76 is 2 x 2 x 19.

What's the prime factorization of 20? ›

What is the prime factorization of 20? The prime factorization of 20 is 2 × 2 × 5 or 22 × 5.

What is the prime factorization of 333? ›

Therefore, the prime factorization of 333 is 3 × 3 × 37.

What is prime factorization of 350? ›

∴ Prime factorization of 350 = 2×5×5×7.

What is the prime factorization of 418? ›

The Prime Factors of 418 are 2 × 11 × 19 and its Factors in Pairs are (1, 418), (2, 209), (11, 38), (19, 22).

What is prime factorization of 125? ›

What are the prime factors of 125? The prime factorisation of 125 is equal 5 x 5 x 5 = 53. Therefore, 5 is the prime factor of 125.

What is the prime factorization of 25 and 45? ›

Prime factorization of 25 and 45 is (5 × 5) and (3 × 3 × 5) respectively. As visible, 25 and 45 have only one common prime factor i.e. 5.

What is the prime factorization of 16? ›

The prime factors of 16 as a product can be written as 2 × 2 × 2 x 2 or 2^4, where 2 is the prime number.

What is prime factorization of 39? ›

There are two prime factors for 39; and they are 3 and 13.

What's the prime factorization of 46? ›

Thus, the prime factorisation of 46 can be written as 2 × 23.

What's the prime factorization of 96? ›

Here the prime numbers multiplied are known as the prime factors of the original number. So the prime factors of 96 are 2 and 3.

What is the prime factorization of 23? ›

Hence, the factors of 23 are 1 and 23. Therefore, 23 is a prime number as it has only 2 factors: 1 and itself.

What is the prime factorization of 24 and 36? ›

Common prime factors of 24 and 36 are 2, 2, 3. Therefore HCF will be, HCF (24, 36) = 2 × 2 × 3 = 12.

What is the prime factorization of 26? ›

So, the prime factorization of 26 is 26 = 2 x 13 and the prime factors of 26 are 2 and 13.

What is the prime factorization of 325? ›

Answer and Explanation: The prime factorization of 325 is 5, 5, and 13.

What is the prime factorization of 170? ›

The Prime Factors and Pair Factors of 170 are 2 × 5 × 17 and (1, 170), (2, 85), (5, 34), (10, 17) respectively.

What is the prime factorization of 108? ›

What is the prime factorization of 108? The prime factorization of 108 is 2 × 2 × 3 × 3 × 3 or 22 × 33.

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