Prime Factorization Calculator

Prime factorization is the process of breaking a number down into the set of prime numbers that multiply together to produce it. Every integer greater than 1 is either prime itself or can be written as a unique product of primes — for example, 360 = 2³ × 3² × 5. This unique breakdown is called the number's prime factorization, and it forms the basis for simplifying fractions, finding the GCF and LCM, and many topics in number theory.

The most common approach is the division method: divide the number by the smallest prime that divides it evenly (starting with 2), then keep dividing the result by that same prime until it no longer divides evenly, then move to the next prime (3, 5, 7, and so on) and repeat. Continue until the remaining value is 1. The primes you divided by, with how many times each was used, form the prime factorization. This calculator automates that process using trial division.

The factor tree method finds a prime factorization visually by splitting a number into any two factors, then splitting each of those factors again, and repeating until every branch ends in a prime number. For example, 360 can branch into 36 × 10, then 36 into 6 × 6 and 10 into 2 × 5, continuing until only primes remain at the tips of the tree. No matter which factor pairs you choose along the way, the final set of prime numbers at the branch tips is always the same, thanks to the fundamental theorem of arithmetic.

A prime number is a whole number greater than 1 that has exactly two positive divisors: 1 and itself. The first several primes are 2, 3, 5, and 7 — notice that 2 is the only even prime, since every other even number is divisible by 2 and therefore has a third divisor. Numbers with more than two divisors, such as 4, 6, 8, 9, and 10, are called composite numbers and can be broken down into prime factors. The number 1 is a special case: it is neither prime nor composite because it has only one divisor.

The fundamental theorem of arithmetic states that every integer greater than 1 is either a prime number itself or can be expressed as a product of prime numbers in exactly one way, aside from the order in which the factors are written. This guarantees that no matter how you break a number down — whether by a factor tree, division method, or any other approach — you always arrive at the same set of prime factors and exponents. It is the mathematical reason prime factorization is meaningful and unique.

Once a number is written in prime factorization form as p₁^a × p₂^b × p₃^c and so on, the total number of positive divisors equals (a+1) × (b+1) × (c+1), multiplying one more than each exponent. For example, 360 = 2³ × 3² × 5¹, so the number of divisors is (3+1) × (2+1) × (1+1) = 4 × 3 × 2 = 24. This shortcut avoids having to list every divisor by hand.

Prime factorization is used to simplify fractions to lowest terms, find the greatest common factor (GCF) and least common multiple (LCM) of two or more numbers, and determine whether a number is a perfect square or cube by checking whether its exponents are all even or all divisible by three. It is also foundational to cryptography — modern encryption systems like RSA rely on the fact that multiplying two large primes together is easy, but factoring the resulting large number back into those primes is computationally very hard.