Factoring Calculator

Find all factors and factor pairs of positive or negative integers using trial division.

Find all factors and factor pairs of any positive integer. Shows total count, all divisors, and multiplication pairs.

Enter a positive integer
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Max 10 trillion
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help_outlineHow to Useexpand_more

Finding Factors of a Number

This calculator finds all factors and factor pairs of any positive or negative integer using trial division. Factors are whole numbers that divide evenly into another number with no remainder.

How to Factor Numbers (Trial Division):

  1. Find the square root: Calculate √n and round down to the nearest whole number
  2. Test divisibility: Check if integers from 1 to √n divide evenly into n
  3. Record factor pairs: When n ÷ i has zero remainder, both i and n/i are factors
  4. Complete the list: Continue until you reach the square root

Positive vs Negative Numbers:

  • Positive integers: Shows only positive factors (standard convention)
  • Negative integers: Shows all factor pairs including negative factors
  • Example: Factors of -6 include (1, -6), (-1, 6), (2, -3), (-2, 3)
infoWhat is Factoring Calculator?expand_more

Factors are whole numbers that multiply together to produce another number. If a × b = c, then both a and b are factors of c. You can also think of factors in terms of division: factors of a number divide evenly into that number with zero remainder.

For example, to find the factors of 16, you identify all pairs of numbers that multiply to give 16: 1 × 16, 2 × 8, and 4 × 4. Therefore, the factors of 16 are 1, 2, 4, 8, and 16.

Important: Every integer has at least two factors: 1 and the number itself. Numbers with exactly two factors are called prime numbers. Numbers with more than two factors are called composite numbers.

functionsFormulaexpand_more

Definition of a Factor:

A is a factor of N if:

N ÷ A = whole number (no remainder)

N mod A = 0

Trial Division Method:

1. Calculate s = floor(√n)

2. Test divisibility for i = 1, 2, 3, ..., s

3. If n ÷ i = 0 remainder, then i and n/i are factors

4. List all unique factors found

Factor Pairs:

If A × B = N, then (A, B) is a factor pair of N

For negative numbers, one factor in each pair must be negative

lightbulbExamplesexpand_more

Example 1: Factors of 18

√18 = 4.24, so check 1 through 4

18 ÷ 1 = 18 → pair (1, 18)

18 ÷ 2 = 9 → pair (2, 9)

18 ÷ 3 = 6 → pair (3, 6)

Factors: 1, 2, 3, 6, 9, 18

Example 2: Factors of 36

√36 = 6, check 1 through 6

Pairs: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6)

Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

Example 3: Factors of 48

√48 = 6.93, check 1 through 6

Pairs: (1, 48), (2, 24), (3, 16), (4, 12), (6, 8)

Factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Example 4: Factors of -6

For negative numbers, factor pairs have mixed signs:

Factor pairs: (1, -6), (-1, 6), (2, -3), (-2, 3)

One positive × one negative = negative product

quizFAQexpand_more
What is the difference between factors and multiples?expand_more
Factors divide INTO a number evenly (factors of 12 include 3), while multiples are what you get when you multiply BY a number (multiples of 3 include 12). A number has finite factors but infinite multiples.
Why do we only check up to the square root?expand_more
When checking for factors, if you find a factor i less than √n, then n/i is automatically a factor greater than √n. So checking up to √n finds all factor pairs without missing any or duplicating work.
What about negative factors for positive numbers?expand_more
Mathematically, negative numbers can be factors of positive numbers (e.g., -2 × -3 = 6). However, by convention, we typically list only positive factors for positive integers. If you need negative factors, simply negate each positive factor.
How do I know if a number is prime?expand_more
A prime number has exactly two factors: 1 and itself. If the factoring calculator returns only these two factors, the number is prime. Numbers with more than two factors are composite.
What is a perfect number?expand_more
A perfect number equals the sum of its proper divisors (all factors except itself). For example, 6 = 1 + 2 + 3, and 28 = 1 + 2 + 4 + 7 + 14. Perfect numbers are rare: the first four are 6, 28, 496, and 8128.