Factor Calculator

This factor calculator lists every positive factor (also called a divisor) of any whole number up to 1,000,000, computes the prime factorization, and flags useful properties like whether the number is prime or perfect. It doubles as a divisors calculator, a prime factorization tool, and a quick way to check the factors of numbers that show up in homework, scheduling, or fraction work.

1. Enter a positive integer up to 1,000,000. The default is 360, which has 24 factors and is one of the most divisible numbers of its size.
2. View all factors listed in sorted order. A factor is any integer that divides evenly into the number with a remainder of zero. The count in the header tells you how many factors the number has.
3. See the prime factorization, written as a product of primes with exponents. For 360 the result is 2³ × 3² × 5. Every integer greater than 1 has exactly one such factorization.
4. Check the summary stats for the number of factors, the sum of factors, whether the input is prime, and whether it is a perfect number (a number equal to the sum of its proper divisors).

If you need the greatest common factor (GCF, also called HCF or GCD) of two numbers, run each one through the calculator, list the factors both share, and pick the largest. For the least common multiple (LCM), compare their prime factorizations and take the highest power of every prime that appears. Both tricks are covered in detail in the Formula section below.

Factor math rests on a small set of definitions and one big idea: every whole number has a unique fingerprint made of primes. Once you have that fingerprint, counting factors, finding GCFs, and computing LCMs all become pattern matching.

Definition: What a Factor Is

A factor of n is an integer that divides n evenly, leaving a remainder of 0. So 4 is a factor of 12 because 12 ÷ 4 = 3 with no remainder. 5 is not a factor of 12 because 12 ÷ 5 = 2 remainder 2. Every positive integer has at least two factors, 1 and itself. The words factor and divisor mean the same thing in this context.

Finding All Factors of a Number

You only have to check integers from 1 up to √n. For every i that divides n, both i and n ÷ i are factors, so you get two factors per check. Stop at √n because beyond that point you would just be listing pairs you already found in reverse.

Worked example: factors of 36
√36 = 6, so check i = 1, 2, 3, 4, 5, 6.

  i = 1 → 36 ÷ 1 = 36   pair: (1, 36)
  i = 2 → 36 ÷ 2 = 18   pair: (2, 18)
  i = 3 → 36 ÷ 3 = 12   pair: (3, 12)
  i = 4 → 36 ÷ 4 = 9    pair: (4, 9)
  i = 5 → 36 ÷ 5 = 7.2  skip (not a factor)
  i = 6 → 36 ÷ 6 = 6    pair: (6, 6)

Factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36}  (9 factors)

Factor Pairs

Every factor below √n pairs with exactly one factor above √n. The only exception is a perfect square, where √n pairs with itself (like 6 × 6 = 36 above). That is why perfect squares always have an odd number of factors, and every other number has an even number of factors.

Number of Factors Formula

Once you have the prime factorization, counting factors is a multiplication. If n = p₁^a × p₂^b × p₃^c × ..., then the number of factors equals (a+1)(b+1)(c+1)... You add 1 to each exponent and multiply.

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

Example 2: 100 = 2² × 5²
Count = (2+1)(2+1) = 9 factors.

Example 3: 1,000,000 = 2⁶ × 5⁶
Count = (6+1)(6+1) = 49 factors.

Prime Factorization by Trial Division

Divide out the smallest prime that works, repeat.

  360 ÷ 2 = 180
  180 ÷ 2 = 90
   90 ÷ 2 = 45
   45 ÷ 3 = 15
   15 ÷ 3 = 5
    5 is prime, stop.

360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5

GCD / GCF / HCF: Greatest Common Divisor

The greatest common divisor (also called greatest common factor or highest common factor) of two numbers is the largest integer that divides both. The cleanest way to find it is from their prime factorizations: take every prime they share, raise it to the smaller of the two exponents, and multiply.

Worked example: GCD(48, 36)

48 = 2⁴ × 3¹
36 = 2² × 3²

Shared primes: 2 and 3.
Smallest exponent of 2 is 2 → 2² = 4
Smallest exponent of 3 is 1 → 3¹ = 3

GCD(48, 36) = 4 × 3 = 12

LCM: Least Common Multiple

The least common multiple of two numbers is the smallest positive integer divisible by both. Take every prime that appears in either factorization and raise it to the larger exponent.

Worked example: LCM(6, 8)

6 = 2¹ × 3¹
8 = 2³

Largest exponent of 2 is 3 → 2³ = 8
Largest exponent of 3 is 1 → 3¹ = 3

LCM(6, 8) = 8 × 3 = 24

Shortcut: LCM(a, b) = (a × b) ÷ GCD(a, b)
        = (6 × 8) ÷ 2 = 48 ÷ 2 = 24

Divisibility Rules (for Mental Checks)

• Divisible by 2: the last digit is 0, 2, 4, 6, or 8.
• Divisible by 3: the sum of the digits is divisible by 3. 357 → 3+5+7 = 15, divisible by 3.
• Divisible by 4: the last two digits form a number divisible by 4. 1,316 ends in 16, so yes.
• Divisible by 5: the last digit is 0 or 5.
• Divisible by 6: divisible by both 2 and 3.
• Divisible by 9: the sum of the digits is divisible by 9. 729 → 7+2+9 = 18.
• Divisible by 10: the last digit is 0.
• Divisible by 11: the alternating digit sum is divisible by 11. 2,728 → 2−7+2−8 = −11, so yes.

Quick Reference: Factors of Common Numbers

Finding factors is not just a textbook exercise. Divisors calculator logic drives fraction simplification, GCF/HCF work in algebra, calendar scheduling problems, and the design of measurement systems (which is why a circle has 360 degrees and an hour has 60 minutes, not 100 of either). This section covers the practical side.

What You Use Factors For

• Simplifying fractions: divide numerator and denominator by their GCD. 48/36 → GCD is 12 → 4/3. No more reducing needed.
• Scheduling and cycles: if one event repeats every 6 days and another every 8 days, they coincide every LCM(6, 8) = 24 days. Factory maintenance, class schedules, and traffic light timing use this.
• Grouping and partitioning: 36 students can be split into equal teams of 1, 2, 3, 4, 6, 9, 12, 18, or 36. The factors of the total are your only options for equal groups with no one left over.
• Perfect numbers: 6, 28, 496, and 8,128 each equal the sum of their proper factors. 6 = 1 + 2 + 3. 28 = 1 + 2 + 4 + 7 + 14. Every known perfect number is even, and they are extremely rare.
• Cryptography: RSA encryption depends on the fact that factoring the product of two very large primes is computationally expensive, even though multiplying them is trivial.

Prime vs Composite vs Neither

A prime number has exactly two factors: 1 and itself. A composite number has three or more factors. The number 1 is neither prime nor composite: it only has one factor (itself), so it does not qualify as prime, and it cannot be built from prime factors.

Highly Composite Numbers

A highly composite number has more factors than any smaller positive integer. These numbers turn up everywhere practical counting happens, because they divide cleanly into the most pieces. 12 splits into halves, thirds, quarters, and sixths. 60 splits into all those plus fifths, tenths, twelfths, fifteenths, twentieths, and thirtieths.

Compare that to 100, which has only 9 factors despite being larger than 48 or 60. Decimal is great for arithmetic, but it is a poor base for carving things into equal parts.

Common Mistakes

• Forgetting 1 and n themselves. The factors of 20 are 1, 2, 4, 5, 10, and 20. Both endpoints count.
• Missing repeated prime factors. 12 = 2 × 2 × 3, not 2 × 3. Prime factorization tracks each prime with its exponent, so 12 = 2² × 3.
• Confusing factors with multiples. Factors of 12 are the numbers that divide into 12 (1, 2, 3, 4, 6, 12). Multiples of 12 are the numbers 12 divides into (12, 24, 36, 48, ...). A factor is small, a multiple is big.
• Mixing up GCD and LCM. GCD is the biggest shared factor, LCM is the smallest shared multiple. GCD(6, 8) = 2. LCM(6, 8) = 24.
• Treating 1 as prime. 1 has only one factor, so it does not fit the definition of a prime. This convention keeps the Fundamental Theorem of Arithmetic working.