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Factor Calculator

Factorization is fundamental to number theory and algebra. A factor is any integer that divides evenly into another number without leaving a remainder. Whether you're simplifying fractions, solving equations, or working through polynomial problems, identifying factors quickly saves time and reduces errors. This calculator generates the complete list of factors for any positive integer you enter.

Last updated: May 12, 2026

Creators Mateusz Mucha, BEng

Reviewers Anna Szczepanek and Jasmine J. Mah

433 people find this calculator helpful

What Is a Factor?

A factor (or divisor) is a whole number that divides another number exactly, with no remainder. For example, 3 and 4 are factors of 12 because 12 ÷ 3 = 4 and 12 ÷ 4 = 3.

Every positive integer has at least two factors: 1 and itself. Some numbers have many more. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. You can think of factors as pairs: multiplying them together always gives your original number. So 3 × 4 = 12, 2 × 6 = 12, and 1 × 12 = 12.

While some definitions include negative factors (−2 and −4 are also factors of 8, since −2 × −4 = 8), most practical applications focus on positive factors only.

How to Identify Factors

A number a is a factor of n if dividing n by a leaves no remainder. Mathematically:

n ÷ a = b, where a and b are both whole numbers

n = a × b

n — The number you're finding factors for
a — A potential factor (divisor)
b — The result of dividing n by a

Divisibility Rules for Quick Recognition

Memorising divisibility rules lets you spot factors without a calculator:

2: Numbers ending in 0, 2, 4, 6, or 8 are divisible by 2.
3: If the sum of digits is divisible by 3, so is the number. (For 27: 2 + 7 = 9, which is divisible by 3.)
4: The last two digits form a number divisible by 4. (712: check 12 ÷ 4 = 3.)
5: Numbers ending in 0 or 5 are divisible by 5.
6: The number must be divisible by both 2 and 3.
7: Double the last digit, subtract from the rest, and repeat. For 343: 34 − (2 × 3) = 28, which is divisible by 7.
8: The last three digits form a number divisible by 8.
9: The sum of all digits is divisible by 9. (For 729: 7 + 2 + 9 = 18, which is divisible by 9.)
10: Any number ending in 0 is divisible by 10.

Prime Factorization and Common Factors

Prime factorization breaks a number into its building blocks—the prime numbers that multiply to form it. For 60, the prime factorization is 2 × 2 × 3 × 5, written as 2² × 3 × 5. This helps identify all factors systematically.

When comparing two numbers, their common factors are integers that divide both. For instance, 12 and 18 share factors 1, 2, 3, and 6. The greatest common factor (GCF)—6 in this case—is especially useful for simplifying fractions: 12/18 = 2/3. In algebra, factoring out common terms from polynomials works the same way: 4x³ + 2x² = 2x²(2x + 1).

Common Pitfalls When Finding Factors

Avoid these frequent mistakes when working with factors:

Forgetting that 1 is a factor — Every positive integer has 1 as a factor. Some learners skip it or assume it's obvious. Always include 1 in your final list.
Confusing factors with multiples — A factor divides into a number; a multiple is what you get when multiplying. 2 is a factor of 8, but 16 is a multiple of 8. They're opposite relationships.
Treating 0 as having factors — Zero is divisible by every non-zero number, so it technically has infinitely many factors. Always restrict your work to positive integers greater than 0.
Missing factor pairs near the square root — When listing factors manually, slow down around √n. Factors close to the square root are easy to overlook. For 36, don't miss 6 × 6.

Frequently Asked Questions

Can a negative number have factors?

In elementary mathematics, factors are defined only for positive integers. However, mathematically speaking, negatives can be factors too. For instance, −2 and −4 are factors of 8 because −2 × −4 = 8. Most practical uses—simplifying fractions, algebraic factoring, cryptography—work with positive factors only. If your problem specifies negative factors, include both positive and negative divisors in your answer.

What is the difference between a factor and a prime factor?

Any factor divides your number evenly, but a prime factor is specifically a prime number that divides it. For 12, the factors are 1, 2, 3, 4, 6, and 12. The prime factors are only 2 and 3 (since 4, 6, and 12 are composite). Prime factorization reduces a number to its simplest building blocks: 12 = 2² × 3. This representation is unique for every positive integer and is fundamental to number theory.

How do I find the greatest common factor of two numbers?

List all factors of both numbers, then identify which ones appear in both lists. For 18 (factors: 1, 2, 3, 6, 9, 18) and 24 (factors: 1, 2, 3, 4, 6, 8, 12, 24), the common factors are 1, 2, 3, and 6. The greatest common factor is 6. Alternatively, use prime factorization: 18 = 2 × 3², and 24 = 2³ × 3. Multiply the lowest powers of all shared primes: 2¹ × 3¹ = 6.

Why is 1 not considered a prime number?

By definition, a prime number has exactly two distinct positive divisors: 1 and itself. The number 1 has only one divisor (itself), so it doesn't meet the definition. Excluding 1 as prime ensures that every integer greater than 1 has a unique prime factorization. If 1 were prime, you could write 12 = 2² × 3, or 12 = 1 × 2² × 3, or 12 = 1 × 1 × 2² × 3, and so on—breaking the uniqueness principle that makes prime factorization so useful.

How do I factor a polynomial expression?

Identify the greatest common factor (GCF) of all terms, then divide it out. For 6x² + 9x, the GCF is 3x, so you write 3x(2x + 3). For more complex polynomials, look for patterns: difference of squares (a² − b² = (a+b)(a−b)), trinomials, or grouping. For instance, x² − 9 = (x+3)(x−3). With trinomials like x² + 5x + 6, find two numbers that multiply to 6 and add to 5 (those are 2 and 3), giving (x+2)(x+3). Always verify your answer by expanding.

What are factor pairs, and why do they matter?

Factor pairs are two numbers that multiply to give your original number. For 20, the pairs are (1, 20), (2, 10), and (4, 5). Every positive integer can be expressed as a product of its factor pairs. They're essential for mental arithmetic, fraction simplification, and understanding the structure of numbers. Knowing factor pairs also speeds up finding all factors: once you've identified one factor, you automatically know its pair. This technique is especially useful when factoring large numbers by hand.

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