What is a Factor?
In mathematics, factors are the integers that are multiplied together to find other integers. Essentially, an integer a is a factor of another integer b so long as b can be divided by a with no remainder.
For example, 6 × 5 = 30. In this example, 6 and 5 are factors of 30. The complete list of factors for 30 would be 1, 2, 3, 5, 6, 10, 15, and 30. Factors are crucial when working with fractions (simplifying them) and finding patterns within numbers.
Factor Pairs
Factors can be organized into factor pairs, which are pairs of numbers that multiply together to equal the original number. For 30, the factor pairs are 1 × 30, 2 × 15, 3 × 10, and 5 × 6. Every positive integer has at least one factor pair: 1 and itself.
Key Terms
- Prime NumberHas exactly two factors: 1 and itself
- Composite NumberHas more than two factors
- Prime FactorA factor that is a prime number
Prime Factorization
Prime factorization involves finding the prime numbers that, when multiplied, return the number being addressed. Every integer greater than 1 is either prime itself or can be uniquely factored into a product of prime numbers (Fundamental Theorem of Arithmetic).
Computing prime factorizations is relatively fast for small numbers, but finding the prime factors of extremely large numbers is very difficult. In fact, modern cryptography (like RSA encryption) relies on the fact that prime factorization of very large numbers is computationally tedious!