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Factor Calculator

Find all the factors of any positive integer and see the step-by-step solution.

Your factors and solution steps will appear here.

What Are Factors?

In mathematics, a factor of an integer `n` is any integer that divides `n` without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Factors always come in pairs; if 3 is a factor of 12, then so is 4 (since `3 × 4 = 12`). Understanding factors is a fundamental concept in arithmetic and number theory. It's the basis for more advanced topics like finding the Greatest Common Factor (GCF) and the Least Common Multiple (LCM), and for simplifying fractions.

How to Find the Factors of a Number

A simple and reliable method to find all the factors of a number, and the one this calculator uses, is through trial division:

  1. Start with 1: Begin checking integers starting from 1.
  2. Divide and Check: Divide your number by the integer you're checking. If there is no remainder, then both the integer you're checking and the result of the division are a factor pair.
  3. Continue Until the Square Root: You only need to check integers up to the square root of your number. Once you pass the square root, you will only find the larger numbers of the pairs you've already found. For example, when finding the factors of 100 (whose square root is 10), once you check 10 and find that `10 × 10` is a pair, you don't need to check any further.

Factors vs. Multiples

It's easy to confuse factors and multiples, but the difference is simple:

  • Factors are numbers you can multiply together to get a certain number. The factors of 12 are `1, 2, 3, 4, 6, 12`.
  • Multiples are the result of multiplying a number by an integer. The multiples of 12 are `12, 24, 36, 48, ...`.

Understanding this distinction is crucial when working with our Greatest Common Factor (GCF) Calculator and Least Common Multiple (LCM) Calculator.

Prime Factorization

A related and powerful concept is prime factorization, which is breaking a number down into its prime number building blocks. For example, the prime factorization of 100 is `2 × 2 × 5 × 5` or `2² × 5²`. This is the fundamental "signature" of a number and is used in many advanced mathematical fields, including cryptography. For a more detailed guide on this topic, the SplashLearn website offers a great introduction.