ConstructionCountdownDate & TimeFinanceHealthMathOther

Factor Calculator

Find all factors of any positive integer instantly — enter a number to get its complete factor list, factor pairs, and prime factorization.

What Is a Factor?

A factor (also called a divisor) of a positive integer n is any positive integer that divides n evenly — that is, with no remainder.

For example, the factors of 18 are 1, 2, 3, 6, 9, and 18, because each divides 18 without a remainder:

  • 18 ÷ 1 = 18 ✓
  • 18 ÷ 2 = 9 ✓
  • 18 ÷ 3 = 6 ✓
  • 18 ÷ 6 = 3 ✓
  • 18 ÷ 9 = 2 ✓
  • 18 ÷ 18 = 1 ✓

Every positive integer has at least two factors: 1 and itself. Numbers with exactly two factors are prime numbers.

How to Find All Factors

The most efficient method checks every integer from 1 up to √n:

  1. For each integer i from 1 to √n, check whether n ÷ i leaves no remainder.
  2. If it divides evenly, both i and n ÷ i are factors.
  3. If i = √n exactly (a perfect square), add it only once.

Example: factors of 36

√36 = 6, so check i = 1 through 6:

i36 ÷ iFactors found
1361, 36
2182, 18
3123, 12
494, 9
57.2— (not whole)
666 (once)

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 — 9 factors total.

Factor Pairs

Factor pairs are two numbers that multiply to give the original number. Every factor has exactly one partner:

Factor pairs of 36:

  • 1 × 36
  • 2 × 18
  • 3 × 12
  • 4 × 9
  • 6 × 6

When the two numbers in a pair are equal (here, 6 × 6), the original number is a perfect square.

Prime Factorization

Prime factorization expresses a number as a product of prime numbers only. It is unique for every integer greater than 1 (Fundamental Theorem of Arithmetic).

Method: Divide repeatedly by the smallest prime that divides the number.

Example: prime factorization of 360

360 ÷ 2 = 180 → 180 ÷ 2 = 90 → 90 ÷ 2 = 45 → 45 ÷ 3 = 15 → 15 ÷ 3 = 5 → 5 is prime

360 = 2³ × 3² × 5

Worked Examples

Factors of 100

Factors: 1, 2, 4, 5, 10, 20, 25, 50, 100 (9 factors)
Prime factorization: 100 = 2² × 5²

Factors of 64

Factors: 1, 2, 4, 8, 16, 32, 64 (7 factors)
Prime factorization: 64 = 2⁶

Factors of 97

97 is a prime number — its only factors are 1 and 97.
Prime factorization: 97 = 97

Sources

Frequently Asked Questions

What is a factor of a number?

A factor (also called a divisor) of a number n is any positive integer that divides n evenly, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 — each divides 12 without a remainder.

How do you find all the factors of a number?

To find all factors of n, check every integer from 1 to √n. If i divides n evenly, both i and n/i are factors. For n = 36: check 1 (→ 1 and 36), 2 (→ 2 and 18), 3 (→ 3 and 12), 4 (→ 4 and 9), 6 (→ 6 and 6). Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36.

What are factor pairs?

Factor pairs are two numbers that multiply together to give the original number. For 36, the factor pairs are (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6). Every factor has a corresponding pair partner — the factor pair approach ensures you find all factors systematically.

What is prime factorization?

Prime factorization expresses a number as a product of its prime factors. For example, 36 = 2² × 3². To find it, divide the number repeatedly by the smallest prime that divides it (starting from 2), then by the next prime, and so on until the quotient is 1.

How many factors does a prime number have?

A prime number has exactly 2 factors: 1 and itself. For example, 17 is divisible only by 1 and 17. The number 1 is not prime — it has only one factor (itself) and is considered neither prime nor composite.