Factoring Calculator
Compute all factors, factor pairs, exponential prime factorization, and visual prime factor tree diagrams.
Number to Factor
Prime Decomposition
The Fundamental Theorem of Arithmetic
Every integer greater than 1 is either a prime number itself or can be uniquely represented as the product of prime numbers (up to the order of the factors). For example, 120 can only be constructed from three 2s, one 3, and one 5 (2 × 2 × 2 × 3 × 5 = 23 × 3 × 5).
For related problem solving and complete calculations, you can also explore our LCM & GCF Calculator and Quadratic Formula Calculator.
Properties of Numbers Based on Factors
- Prime Numbers: Numbers that have exactly two factors: 1 and themselves (e.g., 2, 3, 5, 7, 11, 13, 17, 19).
- Composite Numbers: Numbers with three or more positive divisors (e.g., 4, 6, 8, 9, 10, 12).
- Perfect Numbers: Numbers whose proper divisors sum exactly to the number itself (e.g., 6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + 7 + 14).
Frequently Asked Questions
What is the difference between factors and prime factors?
Factors are any whole numbers that divide evenly into a number without leaving a remainder. Prime factors are the subset of factors that are prime numbers (greater than 1 and divisible only by 1 and themselves). The Fundamental Theorem of Arithmetic states that every integer greater than 1 has a unique prime factorization.
How does a prime factor tree work?
A factor tree breaks a composite number down into two factors repeatedly. If a branch reaches a prime number, that branch stops (it is a leaf). When all terminal branches are prime numbers, their product equals the original number.
How do you find the total number of divisors of a number?
Write the prime factorization as p1^a1 × p2^a2 × ... × pk^ak. The total count of divisors d(n) equals (a1 + 1) × (a2 + 1) × ... × (ak + 1).
Why is 1 neither prime nor composite?
By mathematical definition, prime numbers must have exactly two distinct positive divisors (1 and itself). Since 1 only has one positive divisor (1), it is classified as a unit, neither prime nor composite.
How are prime factors used in cryptography?
Modern internet encryption (like RSA) relies on the fact that multiplying two large prime numbers is computationally instantaneous, but finding the prime factors of a huge 2048-bit number takes classical computers millions of years.