Enter any whole number and see its prime factorization in exponent form, every prime factor, every divisor, and whether the number itself is prime, updating instantly as you type.
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Every integer greater than 1 is either a prime number itself, or it can be written as a product of prime numbers in exactly one way (up to the order you multiply them). This is called the Fundamental Theorem of Arithmetic. Prime factorization is the process of finding that unique set of primes. For example, 60 breaks down into 2 × 2 × 3 × 5, written compactly as 2² × 3 × 5.
This uniqueness is what makes prime factorization so useful. No matter which order you divide out the primes, or which two factors you start splitting a number into on a factor tree, you always end up with the exact same set of primes and exponents. That guarantee is what lets prime factorization be used as a reliable building block for other calculations, including finding the greatest common factor and the least common multiple. It also underlies the least common denominator itself, since the LCD of a set of fractions is just the LCM of their denominators.
Two equivalent methods reach the same factorization.
Divide the number by the smallest prime that divides it evenly, then keep dividing the result by the smallest prime again, until you reach 1. For 60: 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1. The primes used were 2, 2, 3, 5, so 60 = 2² × 3 × 5.
Split the number into any two factors, then keep splitting each branch that isn't prime yet. For 60, split into 6 × 10, then split 6 into 2 × 3 and 10 into 2 × 5. Every branch now ends in a prime: 2, 3, 2, 5. Multiplying them gives the same 2² × 3 × 5.
Finding the least common denominator of several fractions comes down to prime factorization of each denominator. Take the highest power of every prime that appears in any denominator, then multiply those powers together.
LCD of 3/4, 4/5, and 2/3 using prime factorization:
A prime number has exactly two positive divisors: 1 and itself. Any number with more than two divisors is composite (it can be built from smaller factors). The number 1 is neither prime nor composite because it has only one divisor. Checking whether a number is prime only requires testing divisibility by primes up to its square root. If none of them divide evenly, the number is prime. That's why 97 is prime: none of 2, 3, 5, or 7 (the primes up to the square root of 97) divide it evenly.
Primes up to 50:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Eight numbers broken down into their prime factors.