Factor Calculator – All Factors of a Number

Find Every Factor, Factor Pair and the Prime Factorization

Factors Formula

Formula

\[ d \text{ is a factor of } n \iff n = d \cdot k \text{ for some whole number } k \] \[ n = p_1^{e_1} \cdot p_2^{e_2} \cdots p_r^{e_r} \;\Rightarrow\; \tau(n) = \prod_{i=1}^{r} (e_i + 1), \qquad \sigma(n) = \prod_{i=1}^{r} \frac{p_i^{e_i + 1} - 1}{p_i - 1} \]

Where:

\(d\) — a factor (divisor) of \(n\)
\(n\) — the number being factored
\(p_i\), \(e_i\) — the distinct prime factors of \(n\) and their exponents in the prime factorization
\(\tau(n)\) — the number of all factors of \(n\)
\(\sigma(n)\) — the sum of all factors of \(n\)

The first line is the definition of a factor: a number \(d\) divides \(n\) exactly, with no remainder. The second line shows how the prime factorization determines both how many factors exist and what they add up to.

Factors – Calculation Example

Example: find all factors of \(n = 360\), the calculator's default value.

\[ 360 = 2^3 \cdot 3^2 \cdot 5 \]

\[ \tau(360) = (3+1)(2+1)(1+1) = 24 \]

\[ \sigma(360) = \frac{2^4 - 1}{2 - 1} \cdot \frac{3^3 - 1}{3 - 1} \cdot \frac{5^2 - 1}{5 - 1} = 15 \cdot 13 \cdot 6 = 1170 \]

Result: 360 has 24 factors: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360. Their sum is 1170. Since the sum of the proper divisors is \(1170 - 360 = 810 > 360\), the number 360 is abundant. The factors form 12 pairs, from \(1 \times 360\) up to \(18 \times 20\).

How to Use the Factor Calculator

Select the Mode: All factors of one number or Common factors of 2–5 numbers.
In All factors mode, type a whole number from 1 to 10¹² into the Number (n) field. A negative sign is allowed; the calculator lists the factors of \(|n|\) and notes the negative ones.
Read the results: All factors in ascending order, Number of factors \(\tau(n)\), Sum of factors \(\sigma(n)\), the Prime factorization, a table of Factor pairs, and the classification as prime, perfect, abundant or deficient.
In Common factors mode, enter 2–5 non-zero whole numbers into the Numbers box, separated by commas, spaces, semicolons or new lines (for example 12, 18, 30). The result lists every factor the numbers share, with the greatest common factor highlighted.

What Factors Are and Why They Matter

A factor of a number \(n\) is any whole number that divides \(n\) without leaving a remainder. For example, 12 is divisible by 1, 2, 3, 4, 6 and 12, so those are all of its factors. Every factor \(d\) of \(n\) comes with a partner: since \(n = d \times k\), the partner is \(k = n \div d\). That is why factors naturally arrange into factor pairs. A number that is a perfect square, such as 36 with the pair \(6 \times 6\), has an odd number of factors because \(\sqrt{n}\) pairs with itself.

The quickest way to find every factor of a large number is the prime factorization. If \(n = p_1^{e_1} \cdot p_2^{e_2} \cdots p_r^{e_r}\), every factor is a product of the same primes with exponents no larger than \(e_i\). The count follows immediately: \(\tau(n) = (e_1+1)(e_2+1)\cdots(e_r+1)\). For step-by-step prime decomposition, see the Prime Factorization Calculator; to test a single candidate, the Prime Number Checker is faster.

The sum of factors \(\sigma(n)\) is more than a curiosity — it classifies the number. If the proper divisors (all factors except \(n\) itself) add up to exactly \(n\), the number is perfect, like 6 or 28. If they add up to more, it is abundant; if less, deficient. Primes are always deficient, since their only proper divisor is 1.

Common factors are the factors two or more numbers share. They are exactly the factors of the greatest common factor (GCF), which is why the calculator finds the GCF first and then lists its divisors. Common factors are the key to simplifying fractions; the companion concept, the least common multiple, is covered by the LCM Calculator.

Assumptions, Limits and Tips

Whole numbers only. Fractions and decimals are rejected. Zero is excluded because it is divisible by every non-zero integer and would produce an infinite list.
Size limit. The absolute value of each number must be at most 10¹² (1,000,000,000,000,000). Very large inputs with thousands of factors are truncated: the list shows the first 2,000 factors and the pair table shows up to 500 pairs, with the remainder counted.
Negative numbers. Entering a negative value lists the positive factors of \(|n|\) and notes that each also has a negative counterpart \(-d\), doubling the total count of integer factors.
Tip. To check quickly whether a candidate divides \(n\) without listing everything, divide \(n\) by it and see whether the quotient is a whole number. Only candidates up to \(\sqrt{n}\) need testing, because any factor larger than \(\sqrt{n}\) is the partner of a smaller one you already found.
Tip. Factor pairs are handy for simplifying radicals and factoring quadratics: if \(a \times b = n\) with a convenient sum or difference, you have found a useful decomposition of \(n\).

Frequently Asked Questions

How do I find all factors of a number by hand?

Test every whole number from 1 up to \(\sqrt{n}\) and divide \(n\) by each candidate. Each time the division leaves no remainder, you have found two factors at once: the candidate \(d\) and its partner \(n \div d\). Example with 60: testing up to \(\sqrt{60} \approx 7.7\) finds the pairs \(1 \times 60\), \(2 \times 30\), \(3 \times 20\), \(4 \times 15\), \(5 \times 12\), \(6 \times 10\). Collect and sort all numbers in the pairs to get the complete list.

Why do perfect squares have an odd number of factors?

Factors normally come in pairs \(d \times (n \div d)\), so the total count is even. For a perfect square like 36, the middle partner is \(\sqrt{36} = 6\), and 6 pairs with itself — \(6 \times 6 = 36\) — so it is counted only once. That single self-pairing makes the total odd. Conversely, if a number has an odd number of factors, it must be a perfect square.

What is the difference between a factor and a multiple?

The two are inverses of each other. A factor of \(n\) divides it exactly, so factors are never larger than \(n\) itself; a multiple of \(n\) is what you get by multiplying, so multiples are never smaller than \(n\). For example, 3 is a factor of 12, and 12 is a multiple of 3. A common confusion: 12 is not a factor of 3, but 3 is also a multiple of 3 — every number is a factor and a multiple of itself.

Is 1 a factor of every number, and is 1 prime?

Yes, 1 divides every whole number exactly, so 1 is always both a factor of \(n\) and paired with \(n\) itself at the top of the factor list. However, 1 is not a prime number: a prime must have exactly two factors, 1 and itself, while 1 has only one. That is also why 1 is neither classified as composite nor as perfect, abundant or deficient.

What is the most factors a number can have?

There is no fixed limit — some numbers are far richer in factors than others of similar size. Such numbers are called highly composite; for example, 720720 packs 240 factors, and 963761198400 (below 10¹²) has 6720 of them. The count grows with the number of distinct prime factors and their exponents: \(\tau(n) = (e_1+1)(e_2+1)\cdots(e_r+1)\), so adding a new prime always multiplies the count.

Do negative numbers count as factors?

In strict integer arithmetic, yes: if \(d\) divides \(n\), so does \(-d\), so −2 and 2 are both divisors of 10, doubling the count. In school usage and most textbook questions, "factors of n" means the positive divisors only, which is what the calculator lists by default. For a negative input, it shows the factors of \(|n|\) and notes the negative counterparts separately.

What are the first perfect numbers, and why are they rare?

A perfect number equals the sum of its proper divisors: the smallest are 6 (1 + 2 + 3), 28 (1 + 2 + 4 + 7 + 14), 496 and 8128. Every known even perfect number has the Euclid–Euler form \(2^{p-1}(2^p - 1)\), where \(2^p - 1\) must be a Mersenne prime, which explains their rarity — only 52 are known, and it is still unproven whether a single odd perfect number exists.