Factor
Find all prime factors, factor pairs, and divisors of any positive integer with step-by-step factorization. Free online factor calculator for number theory and math problems.
About This Calculator
This Factor Calculator finds all prime factors, divisors, and factor pairs of any positive integer up to very large values using trial division up to the square root of the number. It provides the complete prime factorization following the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 has a unique prime factorization.
The calculator shows the prime factors as a multiplication chain (e.g., 60 = 2 x 2 x 3 x 5), all divisors in ascending order, and all factor pairs (e.g., 1x60, 2x30, 3x20, 4x15, 5x12, 6x10). It also indicates whether the input number is prime (only two factors: 1 and itself) or composite (more than two factors).
Understanding factors is essential for arithmetic operations (simplifying fractions, finding LCM and GCD), algebra (factoring polynomials), number theory (primality and divisibility), and real-world applications (arranging items, division of resources, and determining optimal dimensions).
Frequently Asked Questions
What are factors of a number?
Factors of a number are positive integers that divide the number exactly without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
What are prime factors?
Prime factors are factors that are prime numbers (divisible only by 1 and themselves). For example, the prime factorization of 60 is 2 x 2 x 3 x 5 (or 2^2 x 3 x 5).
How do you find the prime factorization of a number?
Start dividing the number by the smallest prime (2) repeatedly until it no longer divides evenly. Then move to the next prime (3, 5, 7, etc.) and continue until the quotient is 1.
What is the difference between a factor and a multiple?
Factors are numbers that divide evenly into the given number (e.g., 5 is a factor of 20). Multiples are numbers that the given number divides evenly into (e.g., 20 is a multiple of 5).
What numbers have exactly two factors?
Prime numbers have exactly two factors: 1 and themselves. For example, 17 has factors 1 and 17. Numbers with more than two factors are called composite numbers.
How are factors used in real life?
Factors are used in arranging items in rows and columns, dividing resources equally, finding common denominators, calculating gear ratios, and determining optimal dimensions in construction and design.