TFT

Factor Calculator – Find All Factors of Any Integer

Find all factors of any integer instantly with our free online factor calculator. Lists every factor in ascending order – perfect for simplifying fractions and solving number theory problems.

Examples:

Understanding Factors

A factor is a whole number that divides another number evenly, leaving no remainder. When you multiply two factors together, you get the original number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these divides 12 without leaving anything left over.

Every number has at least two factors: 1 and itself. Numbers with exactly two factors are called prime numbers. Numbers with more than two factors are called composite numbers. The number 1 is special – it has only one factor and is neither prime nor composite.

How to Find Factors

  1. 1

    Start with 1 and the number itself

    Every number is divisible by 1 and by itself. These are always your first and last factors.

  2. 2

    Test numbers up to the square root

    Check each number from 2 up to the square root of your number. If it divides evenly, both the divisor and the quotient are factors.

  3. 3

    Use divisibility rules

    Even numbers are divisible by 2. Numbers ending in 0 or 5 are divisible by 5. If the digits add up to a multiple of 3, the number is divisible by 3.

  4. 4

    List factor pairs

    Write factors as pairs that multiply to give the original number. This ensures you don't miss any.

Worked Examples

Example 1: Find all factors of 24

Start with 1 and 24: 1 × 24 = 24 ✓
Check 2: 24 ÷ 2 = 12, so 2 × 12 = 24 ✓
Check 3: 24 ÷ 3 = 8, so 3 × 8 = 24 ✓
Check 4: 24 ÷ 4 = 6, so 4 × 6 = 24 ✓
Check 5: 24 ÷ 5 = 4.8 (not a whole number) ✗
√24 ≈ 4.9, so we stop here
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 (8 factors)

Example 2: Find all factors of 36

1 × 36 = 36 ✓
2 × 18 = 36 ✓
3 × 12 = 36 ✓
4 × 9 = 36 ✓
5 doesn't divide 36 ✗
6 × 6 = 36 ✓ (perfect square, 6 pairs with itself)
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 (9 factors)

Example 3: Find all factors of 97

1 × 97 = 97 ✓
Check 2: 97 is odd ✗
Check 3: 9 + 7 = 16, not divisible by 3 ✗
Check 5: doesn't end in 0 or 5 ✗
Check 7: 97 ÷ 7 = 13.86... ✗
√97 ≈ 9.8, check up to 9
Check 8, 9: neither divides evenly ✗
Factors of 97: 1, 97 (only 2 factors – it's prime!)

Example 4: Prime factorization of 60

60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1
60 = 2² × 3 × 5

Quick Fact

The ancient Greek mathematician Euclid proved around 300 BCE that there are infinitely many prime numbers. His elegant proof by contradiction is still taught today and is considered one of the most beautiful arguments in mathematics.

Frequently Asked Questions

What's the difference between factors and multiples?

Factors divide into a number evenly. Multiples are what you get when you multiply a number by integers. For 12: factors are 1, 2, 3, 4, 6, 12 (finite list). Multiples are 12, 24, 36, 48... (infinite list).

How many factors does a number have?

Use the prime factorization. If n = p₁^a × p₂^b × p₃^c..., then the number of factors is (a+1)(b+1)(c+1).... For 60 = 2² × 3¹ × 5¹, factors = (2+1)(1+1)(1+1) = 3 × 2 × 2 = 12 factors.

What are factor pairs?

Factor pairs are two numbers that multiply together to give the original number. For 20: (1, 20), (2, 10), (4, 5). Perfect squares have one factor that pairs with itself, like (6, 6) for 36.

Why do we only check up to the square root?

Factors come in pairs. Once you pass the square root, you're just finding the partners of factors you already discovered. For 36, after checking 6, any new factor would pair with something less than 6, which you've already found.

What's prime factorization used for?

Prime factorization helps find the GCD and LCM of numbers, simplify fractions, and solve problems in cryptography. It's the unique "fingerprint" of a number – every integer has exactly one prime factorization.

Can negative numbers have factors?

Yes, but we typically work with positive factors. Technically, -2 is also a factor of 12 because (-2) × (-6) = 12. This calculator focuses on positive factors, which is standard for most applications.

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