What Are Factors in Math?

A factor of a number is an integer that divides the number evenly, with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers can divide 12 without leaving a remainder. Factors often come in pairs. The factor pairs of 12 are:

• 1 × 12
• 2 × 6
• 3 × 4

Prime Numbers and Prime Factorization

A prime number is a positive integer greater than 1 that has exactly two distinct factors: 1 and itself. Examples include 2, 3, 5, 7, 11, and 13. Our calculator will tell you if your number is prime.

Prime factorization is the process of breaking a number down into its constituent prime numbers. Every composite (non-prime) number can be expressed as a unique product of prime factors. For example, the prime factorization of 12 is 2 × 2 × 3, which can also be written as 2² × 3.

How to Find Factors Manually (Trial Division)

To find the factors of a number, you can use trial division:

1. Start with 1. Every number is divisible by 1.
2. Test each integer (2, 3, 4, ...) to see if it divides your number evenly.
3. If an integer i divides your number n, then both i and n/i are factors.
4. You only need to test up to the square root of your number. Once you pass the square root, you will just be finding the other half of the factor pairs you've already discovered.

Applications of Factoring

Factoring is a cornerstone of number theory and has important applications:

• Simplifying Fractions: To simplify a fraction, you find the Greatest Common Factor (GCF) of the numerator and denominator, which requires knowing their factors.
• Finding a Common Denominator: When adding or subtracting fractions, you need to find the Least Common Multiple (LCM) of the denominators, a process that is closely related to prime factorization.
• Cryptography: The security of many modern encryption systems, like RSA, relies on the fact that it is computationally very difficult to find the prime factors of extremely large numbers.

Related Math Tools

If you're working on number theory, these calculators might also be helpful:

• Math Calculators - Our full suite of mathematical tools.
• Greatest Common Factor (GCF) Calculator
• Least Common Multiple (LCM) Calculator