Factoring Calculator
The Factoring Calculator helps you find all the factors and factor pairs of any positive or negative integer. Simply enter a whole number, and the calculator will list all the numbers that divide evenly into it. For positive numbers, you’ll get only positive factors (since that’s the standard way to present them), while for negative numbers, both negative and positive factor pairs are shown.
Explore more tools like this in the Math Calculators section of CalculatorCave, where you’ll find dozens of helpful math utilities for everyday calculations.
What Is a Factor?
A factor is a whole number that divides another number evenly, leaving no remainder. In multiplication terms, if a × b = c, then both a and b are factors of c.
For instance:
- 2 × 3 = 6, so both 2 and 3 are factors of 6.
- 4 × 4 = 16, so 4 is a factor of 16.
- 1 × 10 = 10, which means 1 and 10 are also factors of 10.
To find the factors of a number, you simply determine all pairs of integers that multiply to produce that number.
How to Find Factors
The Factoring Calculator uses a straightforward method called trial division. This process checks each integer up to the square root of your number to see whether it divides evenly. Here’s how it works:
- Find the square root of the number n and round it down to the nearest whole number (call this s).
- Start with 1. Since n ÷ 1 = n, the pair (1, n) is your first factor pair.
- Continue dividing n by 2, 3, 4 … up to s. Whenever the result is a whole number, both the divisor and quotient are factors.
- When you’ve tested all numbers up to s, you’ll have a complete list of factors.
Example:
To find the factors of 18, first take its square root:
√18 = 4.24 → round down to 4.
Now test integers from 1 to 4:
- 18 ÷ 1 = 18 → factors (1, 18)
- 18 ÷ 2 = 9 → factors (2, 9)
- 18 ÷ 3 = 6 → factors (3, 6)
Hence, the factors of 18 are 1, 2, 3, 6, 9, 18.
You can also check your results using tools like the Expanded Form Calculator, which helps visualize how numbers are built from their components.
Example Factor Lists
Here’s a quick reference table of common numbers and their factors:
| Number | Factors |
|---|---|
| 3 | 1, 3 |
| 18 | 1, 2, 3, 6, 9, 18 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 |
Notice that every integer has at least two factors — 1 and the number itself. If a number has exactly two factors, it’s called a prime number. Examples include 2, 3, 5, 7, and 11.
Factors of Negative Numbers
Factoring negative numbers follows the same rules, except we include both positive and negative pairs. For instance, the factors of –6 are:
(1, –6), (–1, 6), (2, –3), and (–2, 3).
The logic is straightforward: multiplying a negative number by a positive number gives a negative product. So, for every positive factor pair of a number, you can generate corresponding negative pairs if the original number is negative.
For calculations involving negative or fractional values, you might also find the Scientific Notation Calculator helpful, especially when working with very large or small numbers.
Understanding Factorization
Factorization means breaking down a number into all of its factors or into its prime components. For example:
16 = 2 × 2 × 2 × 2
Here, 16’s prime factors are all 2’s.
36 = 2 × 2 × 3 × 3
So, 36’s prime factors are 2 and 3.
This process helps simplify fractions, find least common multiples (LCM), and greatest common divisors (GCD).
Key Takeaways
- Every number has at least two factors: 1 and itself.
- Factors multiply together to produce the original number.
- Prime numbers have exactly two factors.
- Negative numbers have both positive and negative factor pairs.
- Factorization helps in simplifying mathematical problems, solving equations, and identifying numerical patterns.
Use the Factoring Calculator to save time when breaking down numbers into their components — whether for homework, data analysis, or quick number checks.
To explore more math tools for deeper number understanding, visit the full collection at CalculatorCave Math Calculators.