Prime Factorization Calculator

Enter the whole number you want to factor. You get the product of primes, the exponent form, the ladder method steps and the number of factors, all at once.

Number to factor

Enter a whole number from 2 up to 100 trillion (100,000,000,000,000).

Result

Enter a whole number in the field on the left and press "Calculate". The result will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formula
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter a whole number of 2 or more and get its prime factorization on the spot, written as a product such as \(2 \times 2 \times 5 \times 5\)
The exponent form (\(2^{2} \times 5^{2}\)) and the steps of the ladder method (upside-down division) are shown too
As an application of prime factorization, it also counts how many factors the number has
Enter a prime number (a number that cannot be broken down further) and it tells you it is prime, so it also works as a prime number checker
A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page

Whole numbers from 2 up to 100 trillion (100,000,000,000,000) can be used. 1 is a special number that is neither prime nor composite, so it has no prime factorization.

What is this calculation used for?

Simplifying fractions by finding the GCF (school and everyday math)

Even a large fraction such as \(\frac{84}{126}\) can be simplified step by step with prime factorization. \(84 = 2^{2} \times 3 \times 7\) and \(126 = 2 \times 3^{2} \times 7\), so the shared part \(2 \times 3 \times 7 = 42\) is the greatest common factor. Divide both by 42 and you get \(\frac{2}{3}\).
Simplifying fractions, finding common denominators and finding the least common multiple are all built on prime factorization, so it makes every calculation with fractions easier to see through.

RSA encryption, which protects online shopping and banking

RSA encryption, which protects things like credit card numbers, relies on the fact that factoring a very large number takes an enormous amount of time, even for computers. In fact, factoring a 232-digit number (RSA-768) took about two years using hundreds of computers.
A computer can factor a number around 100 trillion (15 digits) in an instant, but the difficulty explodes as the number of digits grows. This gap is what keeps the internet secure.

Designing gears and repeating cycles (manufacturing)

A gear with 12 teeth and a gear with 18 teeth return to their starting position after 36 teeth have passed, the least common multiple of \(12 = 2^{2} \times 3\) and \(18 = 2 \times 3^{2}\) (3 turns of the small gear and 2 turns of the large one).
Prime factorization is a reliable tool for finding least common multiples, so it is used in designs that spread gear wear evenly and in working out when several cycles line up.

Counting the ways to split a group evenly (teams and classrooms)

How many ways can 36 people be split into groups of the same size? From \(36 = 2^{2} \times 3^{2}\), the number of factors is \((2+1) \times (2+1) = 9\), so there are 9 ways, from "36 groups of 1" to "1 group of 36".
For teams, tiling or packing products into boxes, whenever you want things to divide evenly, you can count the options without listing them all.

Periodical cicadas, prime numbers in nature

In the eastern United States, some cicadas come out of the ground all at once every 13 or 17 years (periodical cicadas). A prime cycle rarely lines up with other cycles. For example, 13-year cicadas and a predator with a 12-year cycle appear in the same year only once every 156 years, their least common multiple.
A leading hypothesis is that having a prime cycle helped them survive. By working out how rarely the cycles line up with prime factorization, you can check this idea for yourself.

Formula

Prime factorization (breaking a number into a product of primes)

Standard notation (the usual math form)

\(n\) \(=\) \(p_1 \times p_2 \times \cdots \times p_k\)

In words (symbols replaced with words)

① \(n\): original number \(=\) ② product of the prime factors \(p_1, p_2, \ldots, p_k\)

The formula in words

① The \(n\): original number can be written as a

② product of the prime factors \(p_1, p_2, \ldots, p_k\) in exactly one way, apart from the order of the factors

Quick example

Breaking down 100 with the ladder method, dividing by the smallest primes first (100 ÷ 2 = 50, 50 ÷ 2 = 25, 25 ÷ 5 = 5), gives

original number (100) \(=\) product of prime factors (2 × 2 × 5 × 5)

\(100 = 2 \times 2 \times 5 \times 5\)

Key idea

Prime numbers (2, 3, 5, 7, 11, …) have no factors other than 1 and themselves. They are the building blocks of the whole numbers, the parts that cannot be split any further. Prime factorization breaks a whole number down into these building blocks, and for every whole number of 2 or more the result is the same every time, apart from the order of the factors (the fundamental theorem of arithmetic). When you factor by hand, the surest way is to divide by the smallest primes in order (2 → 3 → 5 → …), each as many times as it goes in evenly (trial division). A factor tree, which splits the number into any two factors and keeps splitting, gives the same primes in the end.

Exponent form (grouping the same prime factors)

Standard notation (the usual math form)

\(n\) \(=\) \(p\) \(a\) \(\times\) \(q\) \(b\) \(\times \cdots\)

In words (symbols replaced with words)

⑤ \(n\): original number \(=\) ① \(p\): first prime factor ② \(a\): how many times \(\times\) ③ \(q\): second prime factor ④ \(b\): how many times \(\times \cdots\)

The formula in words

① Multiply the \(p\): first prime factor

② by itself \(a\): how many times (exponent)

③ multiply the \(q\): second prime factor

④ by itself \(b\): how many times (exponent) and multiply everything together

⑤ to get back the \(n\): original number

Quick example

Grouping the same prime factors in 100 = 2 × 2 × 5 × 5 and writing them in exponent form gives

original number (100) \(=\) prime factor 2 2 times \(\times\) prime factor 5 2 times

\(100 = 2 \times 2 \times 5 \times 5 = 2^{2} \times 5^{2}\)

Key idea

When the same prime factor appears 2 or more times, group it by writing a small raised number (the exponent) that shows how many times it is multiplied. This is called exponent form, or exponential form. Write the prime factors from smallest to largest. A prime factor that appears only once gets no exponent. For example, in \(360 = 2^{3} \times 3^{2} \times 5\), the 5 is written as is, without an exponent.

Number of factors (an application of prime factorization)

Standard notation (the usual math form)

\(d\) \(=\) \((a+1)\) \(\times\) \((b+1)\)

In words (symbols replaced with words)

③ \(d\): number of factors \(=\) ① \((a+1)\): exponent of \(p\) plus 1 \(\times\) ② \((b+1)\): exponent of \(q\) plus 1

The formula in words

① Multiply the \((a+1)\): exponent of the prime factor \(p\) plus 1

② by the \((b+1)\): exponent of the prime factor \(q\) plus 1

③ and you get the \(d\): number of factors

Quick example

The prime factorization of 100 is \(2^{2} \times 5^{2}\). So the number of factors of 100 is

\(d\): number of factors \(=\) exponent of 2 plus 1 (2 + 1) \(\times\) exponent of 5 plus 1 (2 + 1)

\((2+1) \times (2+1) = 3 \times 3 = 9\)

Key idea

If you actually list the factors of 100, you get 1, 2, 4, 5, 10, 20, 25, 50 and 100: 9 factors, the same as the formula. Why add 1 to each exponent? Every factor of 100 is made by choosing the prime factor 2 zero, one or two times (3 choices) and the prime factor 5 zero, one or two times (3 choices), and multiplying. Choosing a prime zero times also counts, so there is one more choice than the exponent. With 3 or more different prime factors, multiply all the (exponent + 1) values in the same way.

Prime factorization breaks a whole number into a product of primes only, and there is exactly one result. When you work it out by hand, the ladder method, dividing by the smallest primes in order, is the surest way.

Symbols and terms

Symbols

\(n\)	n	The original whole number you want to factor (2 or more). (Example - 100)
\(p, q\)	p, q	Prime factors (primes that divide \(n\) evenly). They are written from smallest to largest. (Example - the prime factors of 100 are 2 and 5)
\(p^{a}\)	p to the a	The prime factor \(p\) multiplied by itself \(a\) times. The small raised \(a\) is the exponent, which tells you to multiply \(a\) times. (Example - \(2^{2} = 2 \times 2 = 4\))
\(d\)	d	The number of factors. The letter comes from "divisor", another word for factor. (Example - for 100, \(d = 9\))

Terms

prime number	A whole number of 2 or more whose only factors are 1 and itself - 2, 3, 5, 7, 11, 13, … 1 is not a prime number, and 2 is the only even prime number.
composite number	A whole number of 2 or more that is not prime. In other words, a number that can be made by multiplying 2 or more primes (example - 100 = 2 × 2 × 5 × 5).
prime factor	A prime number that divides a whole number evenly. The prime factors of 100 are 2 and 5.
prime factorization	Breaking a whole number into a product of primes only. The result is always the same, apart from the order of the factors.
factor (divisor)	A whole number that divides a number evenly. 12 has 6 factors - 1, 2, 3, 4, 6 and 12.
exponent	The small raised number that tells how many times to multiply. \(2^{3}\) is 2 multiplied 3 times (\(2 \times 2 \times 2 = 8\)).
fundamental theorem of arithmetic	The theorem that every whole number of 2 or more can be written as a product of primes in exactly one way. It is the reason a prime factorization has only one answer.
ladder method	A written method that keeps dividing by small primes, also called upside-down division. Write the prime you divide by on the left and the quotient below, and stop when the quotient is a prime. Multiplying all the primes on the left and the last quotient gives back the original number. Many US classrooms also use a factor tree, which reaches the same primes.
trial division	A way to factor a number by testing whether it divides evenly by the smallest primes in order - 2, 3, 5, … If nothing up to \(\sqrt{n}\) divides it evenly, the number that is left is prime.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the quickest way forward.

Multiplication facts and division (Grade 3)	Being able to divide a 2-digit number by a 1-digit number and tell whether it divides evenly
Even and odd numbers, factors and multiples (Grade 4)	Knowing what factors (numbers that divide a number evenly) and multiples are Divisibility rules speed things up - a number is divisible by 2 if its ones digit is even, and by 3 if the sum of its digits is a multiple of 3
Prime numbers and prime factorization (Grades 4–6)	Knowing that a prime number is a whole number of 2 or more with no factors other than 1 and itself (1 is not prime) Knowing that dividing by the smallest primes in order breaks any whole number into a product of primes
Powers and exponents (Grade 6)	Knowing that the small raised number (the exponent) tells how many times to multiply, as in \(2^{3} = 2 \times 2 \times 2\)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to check that the prime factors multiply back to the original number

Prime factor 1	2
Prime factor 2	2
Prime factor 3	5
Prime factor 4	5
Original number n	=B1B2B3*B4

Table to get the original number from the exponent form

Prime factor p	2
Exponent of p, a	2
Prime factor q	5
Exponent of q, b	2
Original number n	=B1^B2*B3^B4

Table to find the number of factors

Exponent of p, a	2
Exponent of q, b	2
Number of factors d	=(B1+1)*(B2+1)

Excel has no function that does prime factorization in one step, so these tables check whether a factorization is correct. Do the factoring itself with the calculator on this page or with Python.
The first table checks that multiplying all the prime factors gives back the original number. B5 shows 2 × 2 × 5 × 5 = 100.
The second table works from the exponent form (2² × 5²). "^" is the symbol for a power (how many times to multiply). B5 shows 100.
The third table finds the number of factors. B3 shows (2+1) × (2+1) = 9. Just replace the prime factors and exponents with your own results.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to check that the prime factors multiply back to the original number

Prime factor 1	2
Prime factor 2	2
Prime factor 3	5
Prime factor 4	5
Original number n	=B1B2B3*B4

Table to get the original number from the exponent form

Prime factor p	2
Exponent of p, a	2
Prime factor q	5
Exponent of q, b	2
Original number n	=B1^B2*B3^B4

Table to find the number of factors

Exponent of p, a	2
Exponent of q, b	2
Number of factors d	=(B1+1)*(B2+1)

The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the prime factors and exponents with your own results.

How to calculate it in Python

number = 100  # the whole number to factor (2 or more)

prime_factors = []   # list of prime factors, smallest first
remaining = number
divisor = 2
while divisor * divisor <= remaining:
    while remaining % divisor == 0:   # keep dividing by the same number while it divides evenly
        prime_factors.append(divisor)
        remaining //= divisor
    divisor += 1
if remaining > 1:                     # whatever is left above 1 is a prime
    prime_factors.append(remaining)

print(f"Prime factors of {number}: {prime_factors}")

Runs with the standard library only. It is trial division written as code - testing whether each number, from the smallest up, divides evenly. "%" gives the remainder of a division, and "//=" divides and drops the decimal part. Change number at the top and run it. For 100 it prints [2, 2, 5, 5].

How to write it in LaTeX and other math languages (copy and paste)

Prime factorization (breaking a number into a product of primes)

n = p₁ × p₂ × ⋯ × pₖ

n = p_1 \times p_2 \times \cdots \times p_k

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>n</mi>
    <mo>=</mo>
    <msub><mi>p</mi><mn>1</mn></msub>
    <mo>&#xD7;</mo>
    <msub><mi>p</mi><mn>2</mn></msub>
    <mo>&#xD7;</mo>
    <mo>&#x22EF;</mo>
    <mo>&#xD7;</mo>
    <msub><mi>p</mi><mi>k</mi></msub>
  </mrow>
</math>

n = p_1 xx p_2 xx cdots xx p_k

FactorInteger[n]

ifactor(n);

factor(n)

n = p_1 × p_2 × ⋯ × p_k

Exponent form (grouping the same prime factors)

n = pᵃ × qᵇ × ⋯

n = p^{a} \times q^{b} \times \cdots

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>n</mi>
    <mo>=</mo>
    <msup><mi>p</mi><mi>a</mi></msup>
    <mo>&#xD7;</mo>
    <msup><mi>q</mi><mi>b</mi></msup>
    <mo>&#xD7;</mo>
    <mo>&#x22EF;</mo>
  </mrow>
</math>

n = p^a xx q^b xx cdots

p^a * q^b

n := p^a * q^b;

n = p^a * q^b;

n = p^a × q^b × ⋯

Number of factors (an application of prime factorization)

d = (a + 1) × (b + 1)

d = (a+1) \times (b+1)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>d</mi>
    <mo>=</mo>
    <mo>(</mo><mi>a</mi><mo>+</mo><mn>1</mn><mo>)</mo>
    <mo>&#xD7;</mo>
    <mo>(</mo><mi>b</mi><mo>+</mo><mn>1</mn><mo>)</mo>
  </mrow>
</math>

d = (a + 1) xx (b + 1)

(a + 1)*(b + 1)

d := (a + 1)*(b + 1);

d = (a + 1)*(b + 1);

d = (a + 1) × (b + 1)

How to have ChatGPT do the calculation

You are a calculation assistant for whole numbers. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

Find the prime factorization of 360.
1. List all the prime factors from smallest to largest (repeat a prime factor as many times as it appears)
2. Write it in exponent form (for example, in a form like 2^3 × 3^2 × 5)
3. Also find the number of factors of 360

Show the code you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula