Factor Calculator

Enter any positive integer to find all its factors, factor pairs, prime factorization, and whether it is prime. Step-by-step trial division work is shown.

What Are Factors?

A factor of a whole number n is any positive integer that divides n exactly, with no remainder. Factors always occur in pairs: if a divides n, then n/a also divides n. This is why factors are listed as pairs (1 × n, 2 × n/2, etc.).

The factors of 360, for example, are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 — a total of 24 factors. This makes 360 a highly composite number, which is why it was historically chosen for the number of degrees in a circle and the number of days in ancient calendar systems.

Finding Factors: Trial Division

The most straightforward method for finding all factors is trial division. You test each integer from 1 up to the square root of n. If a number divides n evenly, both the divisor and the quotient are factors.

You only need to check up to √n because factor pairs straddle the square root. If a × b = n and a ≤ b, then a ≤ √n ≤ b. Every factor below the square root has a corresponding factor above it.

Example: Find factors of 36
√36 = 6. Check divisors 1 through 6:
36 ÷ 1 = 36 ✓   36 ÷ 2 = 18 ✓   36 ÷ 3 = 12 ✓   36 ÷ 4 = 9 ✓   36 ÷ 5 = 7.2 ×   36 ÷ 6 = 6 ✓
Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

Prime vs. Composite Numbers

Every positive integer falls into one of three categories:

• 1 — neither prime nor composite. It has only one factor (itself).
• Prime — has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
• Composite — has three or more factors. Can be written as a product of smaller primes. Examples: 4 = 2², 15 = 3 × 5, 100 = 2² × 5².

The Fundamental Theorem of Arithmetic states that every integer greater than 1 has a unique prime factorization. This is why prime numbers are called the "building blocks" of all integers.

Prime Factorization

Prime factorization decomposes a number into a product of prime numbers. The result is written using exponential notation when primes repeat: 360 = 2³ × 3² × 5.

Uses of prime factorization:

• Finding GCF: Multiply all prime factors that appear in both numbers, using the smaller exponent. GCF(36, 48) = GCF(2²×3², 2⁴×3) = 2²×3 = 12.
• Finding LCM: Multiply all prime factors that appear in either number, using the larger exponent. LCM(36, 48) = 2⁴×3² = 144.
• Simplifying fractions: Factor the numerator and denominator, then cancel common prime factors.
• Cryptography: RSA encryption security relies on the computational difficulty of factoring very large numbers (hundreds of digits) into primes.

Factor Trees

A factor tree is a visual way to find the prime factorization of a number. Start with the number at the top. Draw two branches to any two factors. If a factor is composite, branch it again. Continue until all branches end in prime numbers. The primes at the leaves form the prime factorization.

Factor tree for 60:

       60
      /  \
     6    10
    / \   / \
   2   3  2   5
  

60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

You always get the same prime factorization regardless of which pair of factors you start with — this is guaranteed by the Fundamental Theorem of Arithmetic.

Counting Factors from Prime Factorization

If you know the prime factorization, you can count the total number of factors without listing them all. For n = p₁^a₁ × p₂^a₂ × … × p_k^a_k:

Number of factors = (a₁ + 1) × (a₂ + 1) × … × (a_k + 1)

Example: 360 = 2³ × 3² × 5¹. Number of factors = (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24.

This formula works because each factor is formed by independently choosing an exponent for each prime from 0 up to the maximum: 2^0,1,2,3 gives 4 choices; 3^0,1,2 gives 3 choices; 5^0,1 gives 2 choices. 4 × 3 × 2 = 24 combinations.

Frequently Asked Questions

Related Calculators

• GCF Calculator — greatest common factor with Euclidean algorithm steps
• LCM Calculator — least common multiple with prime factorization method