TFT

Factors List Generator – Find All Factors of a Number

Generate a complete sorted list of all factors of any number instantly with our free online factors calculator. Ideal for math homework, LCM/GCD problems, and number theory.

Examples:

Understanding Factors

Factors are the building blocks of numbers. When you break down a number into its factors, you're finding all the whole numbers that divide into it evenly. This is different from prime factorization, which breaks a number down into only prime numbers.

The sum of all factors (excluding the number itself) tells you something interesting about the number. If the sum equals the number, it's a perfect number like 6 or 28. If the sum is greater, it's abundant. If less, it's deficient.

How to Generate a Factors List

  1. 1

    Start with 1

    Every number has 1 as a factor. Write it down first.

  2. 2

    Work through each number

    Test 2, 3, 4, and so on. If the number divides evenly, write down both the divisor and the quotient.

  3. 3

    Stop at the square root

    Once you reach the square root, you've found all unique factor pairs. Any further testing just repeats what you already have.

  4. 4

    Sort the list

    Arrange all factors from smallest to largest for a clean, organized list.

Worked Examples

Example 1: Factors of 28

1 × 28 = 28 ✓
2 × 14 = 28 ✓
4 × 7 = 28 ✓
√28 ≈ 5.3, check up to 5
3, 5 don't divide evenly ✗
Factors: 1, 2, 4, 7, 14, 28
Sum: 1 + 2 + 4 + 7 + 14 + 28 = 56
Note: 1 + 2 + 4 + 7 + 14 = 28, so 28 is a perfect number!

Example 2: Factors of 45

1 × 45 = 45 ✓
3 × 15 = 45 ✓ (4 + 5 = 9, divisible by 3)
5 × 9 = 45 ✓ (ends in 5)
√45 ≈ 6.7, check up to 6
2, 4, 6 don't divide (45 is odd) ✗
Factors: 1, 3, 5, 9, 15, 45
Sum: 1 + 3 + 5 + 9 + 15 + 45 = 78

Example 3: Factors of 64

1 × 64 = 64 ✓
2 × 32 = 64 ✓
4 × 16 = 64 ✓
8 × 8 = 64 ✓ (perfect square!)
Factors: 1, 2, 4, 8, 16, 32, 64
Sum: 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127
Note: 64 = 2⁶, so it has 6 + 1 = 7 factors

Example 4: Factors of 100

1 × 100 = 100 ✓
2 × 50 = 100 ✓
4 × 25 = 100 ✓
5 × 20 = 100 ✓
10 × 10 = 100 ✓ (perfect square)
Factors: 1, 2, 4, 5, 10, 20, 25, 50, 100
Sum: 217

Quick Fact

Perfect numbers are incredibly rare. The first four are 6, 28, 496, and 8128. The ancient Greeks knew these four. The fifth perfect number, 33,550,336, wasn't discovered until the 15th century. Today, only 51 perfect numbers are known, all discovered using computers.

Frequently Asked Questions

What's the fastest way to find all factors?

Use divisibility rules first. Check 2 (even numbers), 3 (digit sum), 5 (ends in 0 or 5), then test remaining numbers up to the square root. This eliminates many candidates quickly.

How do I know if I found all factors?

If you've tested every number up to the square root and recorded both the divisor and quotient for each successful division, you have them all. The square root itself is a factor only for perfect squares.

What does the sum of factors tell me?

Compare the sum of proper factors (excluding the number itself) to the number. Equal means perfect (like 6). Greater means abundant (like 12). Less means deficient (like 8).

Why are factor pairs useful?

Factor pairs help visualize multiplication and division relationships. They're essential for factoring polynomials, simplifying fractions, and understanding number structure.

Can decimals have factors?

Factors are defined for integers only. While you can divide decimals, the concept of "factors" specifically refers to whole number divisors that produce whole number quotients.

What's a proper factor?

Proper factors exclude the number itself. For 12, proper factors are 1, 2, 3, 4, 6. Some definitions also exclude 1, giving just 2, 3, 4, 6. Context determines which definition applies.

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