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Factor Tree & Division Method
Break down any number into its prime factors. View results in exponential form, expanded form, and see step-by-step solutions using the division method with factor tree visualization.
Enter a positive integer greater than 1
Enter a number to see its prime factorization
Prime factorization is the process of breaking down a composite number into a product of prime numbers. Every composite number has a unique prime factorization (Fundamental Theorem of Arithmetic).
Exponential form (like 2³ × 3²) is more compact than expanded form (2 × 2 × 2 × 3 × 3).
Every positive integer greater than 1 has a unique prime factorization.
Start with any factor pair and keep breaking down until all branches end in primes.
Divide repeatedly by the smallest prime until you reach 1. List all divisors used.
The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either prime or can be uniquely represented as a product of prime numbers (up to the order of factors). This means 60 can only be factored as 2² × 3 × 5, regardless of which method you use or in what order you find the factors.
Start by testing small primes (2, 3, 5, 7, 11...). Divide by 2 as many times as possible, then try 3, then 5, and so on. You only need to test primes up to the square root of the number. If a number is still greater than 1 after testing all primes up to its square root, that number itself must be prime.
The number 1 has no prime factorization. By definition, prime factorization involves expressing a number as a product of primes, but 1 is not prime and is not composite. The number 1 is the multiplicative identity and stands alone as a special case.
Prime factorization is fundamental in mathematics. It's used to find GCF and LCM, simplify fractions, solve divisibility problems, in number theory, and even in modern cryptography (like RSA encryption). It reveals the basic structure and building blocks of numbers.
No, this is impossible due to the Fundamental Theorem of Arithmetic. Each number has a unique prime factorization. If two numbers have identical prime factorizations, they must be the same number. This uniqueness property is what makes prime factorization so valuable in mathematics.
Factors are all numbers that divide evenly into a number. Prime factors are specifically the prime numbers that multiply together to give the original number. For example, 12 has factors 1, 2, 3, 4, 6, 12, but its prime factors are only 2, 2, and 3 (or 2² × 3).
Prime factorization reveals the fundamental building blocks of numbers. Just as molecules are made of atoms, all composite numbers are made of prime numbers multiplied together. This concept is one of the most important in number theory and has applications ranging from simplifying fractions to securing internet communications.