Enter one positive integer to find its factors. The list of factors, the number of factors, the factor pairs and the prime factorization are all calculated at once.
Enter a positive integer with 1 to 14 digits (for example, 120). Factors cannot be found for 0, negative numbers or decimals.
Result
Enter a positive integer in the field on the left and press "Calculate". The result will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formula
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter one positive integer, and all of its factors are listed from smallest to largest (none missed, none repeated)
- You also get the factor pairs (pairs that multiply to the number) and the number of factors at the same time
- The prime factorization (in a form like \(120 = 2 \times 2 \times 2 \times 3 \times 5\)) is shown too, with each division step
- If the only factors are 1 and the number itself, it is a prime number, so you can also use this page to check whether a number is prime
- A plain-language explanation of how to find factors and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
This page works with positive integers (1 or more, up to 14 digits). It does not handle factors of 0, negative numbers or decimals (factors here are only about positive integers dividing evenly into each other).
What is this calculation used for?
Simplifying fractions (the foundation for fraction skills)
To simplify \(\frac{24}{36}\), find a factor that the numerator 24 and the denominator 36 share, and divide both by it (divide by their greatest common factor, 12, and you get \(\frac{2}{3}\) in one step).
Once factors come to mind quickly, simplifying fractions, finding common denominators and simplifying ratios all become faster and more accurate. Many students who struggle with fractions are really struggling with this sense of factors.
Planning equal groups (everyday life and parties)
If you want to hand out 120 cookies so that everyone gets the same number with none left over, the possible numbers of people are exactly the factors of 120 (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120).
Splitting a class into teams, filling goody bags, sharing supplies equally: listing the factors gives you every option for "how many people (or bags) can share this evenly".
Rectangular layouts for tiles, store displays and seating
The ways to arrange 36 tiles into a rectangle with none left over are exactly the factor pairs: 1×36, 2×18, 3×12, 4×9 and 6×6, five in all.
Laying out floor or wall tiles, arranging products on a shelf, setting up chairs for a ceremony: whenever a fixed number of items goes into a rectangle, the factor pairs are your layout options.
Encryption that protects online shopping and banking (RSA)
Prime factorization takes an instant for numbers of about 14 digits, but for huge numbers hundreds of digits long, it cannot be finished in any realistic time, even with all the computers in the world. RSA encryption, widely used on the internet, relies on this fact: "big numbers cannot be factored in practice".
When you send your card number to an online store, prime factorization, the same thing this page does, is at the root of what keeps your information safe.
Choosing gear tooth counts (machines, cars and watches)
If the tooth counts of two meshing gears share a factor, the same teeth keep meeting the same partners, so wear and damage tend to build up on certain teeth. That is why machine designers often choose tooth counts that share no factor other than 1 (that are relatively prime).
From car transmissions to wristwatches, whether two numbers share a factor affects how long a product lasts.
Formula
What a factor is (checking with multiplication)
Standard notation
(the usual math form)
\(N\)
\(=\)
\(A\)
\(\times\)
\(B\)
In words
(symbols replaced with words)
③
\(N\): number to factor
\(=\)
①
\(A\): factor
\(\times\)
②
\(B\): its partner factor
The formula in words
①
When the
\(A\): factor
②
times the
\(B\): its partner factor
③
equals exactly the
\(N\): number to factor
, both \(A\) and \(B\) are factors of \(N\)
Quick example
120 can be written as the product 4 × 30, so 4 and 30 are both factors of 120
number to factor (120)
\(=\)
factor (4)
\(\times\)
partner factor (30)
\(120 \div 4 = 30\)
\(4 \times 30 = 120\)
Key idea
To check whether a number is a factor, see whether it divides evenly. 120 ÷ 4 = 30 with no remainder, so 4 is a factor of 120 (and the answer, 30, is automatically a factor too). If there is a remainder, as in 120 ÷ 7 = 17 remainder 1, the number is not a factor.
Factors always come in these "pairs that multiply to \(N\)", so the smaller factor of each pair is always at most \(\sqrt{N}\) (the number that gives \(N\) when squared). So if you try dividing by every number up to \(\sqrt{N}\), you find every factor. When both numbers in a pair are the same, as in 36 = 6 × 6 (a perfect square), 6 is counted only once. That is why a perfect square always has an odd number of factors.
Prime factorization (writing a number as a product of primes)
Standard notation
(the usual math form)
\(N\)
\(=\)
\(p\)
\(a\)
\(\times\)
\(q\)
\(b\)
\(\times\)
\(r\)
\(c\)
In words
(symbols replaced with words)
⑤
\(N\): number to factor
\(=\)
①
\(p\): prime number
②
\(a\): times \(p\) is multiplied
\(\times\)
③
\(q\): another prime
\(b\): times \(q\) is multiplied
\(\times\)
④
\(r\): yet another prime
\(c\): times \(r\) is multiplied
The formula in words
①
Multiply the
\(p\): prime number
②
by itself the number of times given by
\(a\): times \(p\) is multiplied
③
do the same with
\(q\): another prime
④
and
\(r\): yet another prime
, each the right number of times, and multiply everything together
⑤
and you get back exactly the
\(N\): number to factor
Quick example
The prime factorization of 120 (it divides evenly by 2 three times, by 3 once and by 5 once)
number to factor (120)
\(=\)
prime (2)
times (3)
\(\times\)
another prime (3)
times (1)
\(\times\)
yet another prime (5)
times (1)
\(120 \div 2 = 60,\quad 60 \div 2 = 30,\quad 30 \div 2 = 15,\quad 15 \div 3 = 5\)
\(120 = 2 \times 2 \times 2 \times 3 \times 5 = 2^{3} \times 3 \times 5\)
Key idea
To find a prime factorization, just start with the smallest prime and keep dividing as long as it divides evenly. For 120: divide by 2 to get 60, by 2 again to get 30, and by 2 once more to get 15. 15 no longer divides by 2, so move to the next prime, 3, and get 5. 5 is prime, so you are done.
A number may use just one prime or four or more different primes, but the result always has the same shape: a product of primes only. What is more, every number has exactly one prime factorization (apart from the order of the factors). This is called the uniqueness of prime factorization (the Fundamental Theorem of Arithmetic), one of the most important facts about integers.
Formula for the number of factors
Standard notation
(the usual math form)
\(d(N)\)
\(=\)
\((a+1)\)
\(\times\)
\((b+1)\)
\(\times\)
\((c+1)\)
In words
(symbols replaced with words)
④
\(d(N)\): number of factors
\(=\)
①
\(a + 1\): times \(p\) is multiplied, plus 1
\(\times\)
②
\(b + 1\): times \(q\) is multiplied, plus 1
\(\times\)
③
\(c + 1\): times \(r\) is multiplied, plus 1
The formula in words
①
Multiply
\(a + 1\): times \(p\) is multiplied, plus 1
②
by
\(b + 1\): times \(q\) is multiplied, plus 1
③
and by
\(c + 1\): times \(r\) is multiplied, plus 1
(one for each different prime)
④
and you get the
\(d(N)\): number of factors
Quick example
Since \(120 = 2^{3} \times 3 \times 5\) (2 is multiplied 3 times, 3 once and 5 once), the number of factors of 120 is
number of factors \(d(120)\)
\(=\)
times for 2, plus 1 (3 + 1)
\(\times\)
times for 3, plus 1 (1 + 1)
\(\times\)
times for 5, plus 1 (1 + 1)
\((3+1) \times (1+1) \times (1+1) = 4 \times 2 \times 2 = 16\)
Key idea
Why does this formula give the count? Every factor of 120 can be made by multiplying "2 used 0 to 3 times, 3 used 0 or 1 time, and 5 used 0 or 1 time" (for example, 12 = 2 × 2 × 3 uses 2 twice, 3 once and 5 zero times). There are 4 choices for 2, 2 choices for 3 and 2 choices for 5, so there are 4 × 2 × 2 = 16 combinations. That is exactly the number of factors. The key is to add 1 to each count, because "use it 0 times" is also a choice.
This formula comes up often in math contests and on standardized tests. You do not have to write out the factors one by one: once you have the prime factorization, you get the count right away.
The basic way to find factors is to look for pairs that multiply to the number, and dividing by every number up to √N finds them all. Once you have the prime factorization, multiplying "(times each prime is used + 1)" gives the number of factors in one step.
Symbols and terms
Symbols
| \(N\) | N | The positive integer whose factors you want. (Example - 120) |
| \(A,\ B\) | A, B | A pair of factors that multiply to \(N\). (Example - the 4 and 30 in \(120 = 4 \times 30\)) |
| \(p,\ q,\ r\) | p, q, r | The primes in a prime factorization (the prime factors). (Example - the 2, 3 and 5 in \(120 = 2^3 \times 3 \times 5\)) |
| \(a,\ b,\ c\) | a, b, c | How many times each prime is multiplied in the prime factorization (the exponents). In the number-of-factors formula, you add 1 to each and multiply the results. (Example - for \(120 = 2^3 \times 3 \times 5\), 2 is multiplied \(a = 3\) times, and 3 and 5 are each multiplied \(b = c = 1\) time) |
| \(2^{3}\) | 2 to the third power (2 cubed) | 2 multiplied by itself 3 times (\(2 \times 2 \times 2 = 8\)). The small raised number (the exponent) tells how many times to multiply. |
| \(d(N)\) | d of N | The number of factors of \(N\). The d comes from "divisor", another word for factor. (Example - \(d(120) = 16\)) |
| \(\sqrt{N}\) | square root of N | The number that gives \(N\) when squared (the square root). In every factor pair, one factor is at most \(\sqrt{N}\), so checking up to here is enough to find all the factors. |
Terms
| factor | A positive integer that divides a given integer evenly (with no remainder). Example - the factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30, eight in all. 1 and the number itself are always factors of any number. |
| multiple | A number you get by multiplying an integer by a whole number (1 times, 2 times, 3 times and so on). Factors and multiples are two sides of the same coin: "4 is a factor of 120" and "120 is a multiple of 4" say the same thing. |
| prime number | An integer of 2 or more whose only factors are 1 and itself. In order they are 2, 3, 5, 7, 11, 13 and so on, without end. 1 is not a prime number. |
| composite number | An integer of 2 or more that is not prime. It has 3 or more factors and can be broken down into a product of primes. (1 is neither prime nor composite.) |
| prime factor | A prime number that divides a given integer evenly. Example - the prime factors of 120 are 2, 3 and 5, three in all. |
| prime factorization | Writing an integer as a product of prime numbers only. Example - \(120 = 2 \times 2 \times 2 \times 3 \times 5\). There is only one way to do it (apart from the order of the factors). |
| divisor | Another word for factor in this sense. In school math, "factor" is the usual word; in number theory, "divisor" is common, and the d in \(d(N)\) comes from it. This page uses "factor" throughout. (In a division such as \(120 \div 4\), the number you divide by is also called the divisor.) |
| perfect square | A number you get by multiplying an integer by itself. Example - \(36 = 6 \times 6\) and \(3600 = 60 \times 60\). A perfect square always has an odd number of factors (because the pair of equal numbers is counted only once). |
| exponent (power) | A way to write the same number multiplied by itself many times. In \(2^3\), the small raised number (the exponent) tells how many times to multiply. |
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.
| Multiplication facts and division (Grades 3–4) |
|
| Factors and multiples (Grade 4) |
|
| Prime numbers and prime factorization (Grades 4–6) |
|
| Exponents (Grade 6) |
|
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 with division whether a number is a factor
| Number to factor N | 120 |
| Number to test A | 4 |
| Remainder of N ÷ A (0 = it is a factor) | =MOD(B1,B2) |
| Partner factor B (when the remainder is 0) | =B1/B2 |
Table to check a prime factorization
| Prime factor 1 | 2 |
| Prime factor 2 | 2 |
| Prime factor 3 | 2 |
| Prime factor 4 | 3 |
| Prime factor 5 | 5 |
| Product of all (correct if it gives back the number) | =PRODUCT(B1:B5) |
Table to find the number of factors with the formula
| Times prime p is multiplied, a | 3 |
| Times prime q is multiplied, b | 1 |
| Times prime r is multiplied, c | 1 |
| Number of factors d(N) | =(B1+1)*(B2+1)*(B3+1) |
The first table checks for a factor with MOD(B1,B2), "the remainder of B1 divided by B2". If B3 is 0, it is a factor, and B4 shows its partner (30 for 120 and 4). If the remainder is not 0, B4 is not a factor, so ignore it.
The second table checks a prime factorization. PRODUCT(B1:B5) multiplies everything from B1 to B5, so if B6 shows 120, the factorization is correct (for a number with fewer than 5 prime factors, enter 1 in the unused cells).
The third table is the number-of-factors formula. B4 shows (3+1)×(1+1)×(1+1) = 16.
The second table checks a prime factorization. PRODUCT(B1:B5) multiplies everything from B1 to B5, so if B6 shows 120, the factorization is correct (for a number with fewer than 5 prime factors, enter 1 in the unused cells).
The third table is the number-of-factors formula. B4 shows (3+1)×(1+1)×(1+1) = 16.
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 with division whether a number is a factor
| Number to factor N | 120 |
| Number to test A | 4 |
| Remainder of N ÷ A (0 = it is a factor) | =MOD(B1,B2) |
| Partner factor B (when the remainder is 0) | =B1/B2 |
Table to check a prime factorization
| Prime factor 1 | 2 |
| Prime factor 2 | 2 |
| Prime factor 3 | 2 |
| Prime factor 4 | 3 |
| Prime factor 5 | 5 |
| Product of all (correct if it gives back the number) | =PRODUCT(B1:B5) |
Table to find the number of factors with the formula
| Times prime p is multiplied, a | 3 |
| Times prime q is multiplied, b | 1 |
| Times prime r is multiplied, c | 1 |
| Number of factors d(N) | =(B1+1)*(B2+1)*(B3+1) |
The same formulas as in Excel (MOD, PRODUCT) work as is. Copy the whole table, paste it into cell A1, and replace the numbers in column B with your own.
How to calculate it in Python
n = 120 # the positive integer to factor
divisors = [] # list of factors
pairs = [] # factor pairs that multiply to n
i = 1
while i * i <= n: # dividing by numbers up to √n finds them all
if n % i == 0: # remainder 0: i is a factor
divisors.append(i)
if i != n // i: # its partner is a factor too (skip the repeat in a square like 6×6)
divisors.append(n // i)
pairs.append((i, n // i))
i += 1
divisors.sort()
prime_factors = [] # prime factorization (keep dividing by the smallest prime while it divides evenly)
rest = n
p = 2
while p * p <= rest:
while rest % p == 0:
prime_factors.append(p)
rest = rest // p
p += 1
if rest > 1:
prime_factors.append(rest) # whatever is left at the end is a prime factor too
print(f"Factors of {n} ({len(divisors)} in total): {divisors}")
print(f"Factor pairs: {pairs}")
if len(prime_factors) >= 2:
print(f"Prime factorization: {n} = {' × '.join(map(str, prime_factors))}")
elif len(prime_factors) == 1:
print(f"{n} is a prime number")
Runs with the standard library only. The code follows the idea of formula 1, "dividing by numbers up to √n finds every factor". Change n at the top and run it (numbers up to about 14 digits finish quickly).
How to write it in LaTeX and other math languages (copy and paste)
What a factor is (checking with multiplication)
N = A × B
N = A \times B
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>N</mi>
<mo>=</mo>
<mi>A</mi>
<mo>×</mo>
<mi>B</mi>
</mrow>
</math>
N = A xx B
A*B
N := A*B;
N = A*B;
N = A × B
Prime factorization (writing a number as a product of primes)
N = pᵃ × qᵇ × rᶜ × …
N = p^{a} \times q^{b} \times r^{c} \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>×</mo>
<msup><mi>q</mi><mi>b</mi></msup>
<mo>×</mo>
<msup><mi>r</mi><mi>c</mi></msup>
<mo>×</mo>
<mo>⋯</mo>
</mrow>
</math>
N = p^a xx q^b xx r^c xx cdots
p^a*q^b*r^c
N := p^a*q^b*r^c;
N = p^a*q^b*r^c;
N = p^a × q^b × r^c × …
Formula for the number of factors
d(N) = (a + 1)(b + 1)(c + 1) …
d(N) = (a+1)(b+1)(c+1)\cdots
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>d</mi><mo>(</mo><mi>N</mi><mo>)</mo>
<mo>=</mo>
<mo>(</mo><mi>a</mi><mo>+</mo><mn>1</mn><mo>)</mo>
<mo>(</mo><mi>b</mi><mo>+</mo><mn>1</mn><mo>)</mo>
<mo>(</mo><mi>c</mi><mo>+</mo><mn>1</mn><mo>)</mo>
<mo>⋯</mo>
</mrow>
</math>
d(N) = (a+1)(b+1)(c+1) cdots
(a + 1)*(b + 1)*(c + 1)
d := (a + 1)*(b + 1)*(c + 1);
d = (a + 1)*(b + 1)*(c + 1);
d(N) = (a+1)(b+1)(c+1)…
How to have ChatGPT do the calculation
You are a calculation assistant for integers. 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). For 120, find each of the following: 1. All factors, listed from smallest to largest, and how many there are 2. All factor pairs that multiply to 120 3. The prime factorization (in a form like 2 × 2 × 2 × 3 × 5) 4. A check with the number-of-factors formula (add 1 to how many times each prime is multiplied, and multiply the results) that it matches the count in item 1 Show the code you used and the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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