Prime Factorization Calculator: Factor Tree, Steps & Exponential Form
Free prime factorization calculator with factor tree and step-by-step Logic Trace. Find the prime factorization of any number. Exponential form, Factor List, copy results. For students, pre-algebra, and number theory.
What This Calculator Does and Who It's For
- Who it's forStudents and teachers learning how to find prime factorization or building factor trees; anyone searching for prime factorization of 100, prime factorization calculator with steps, or factor tree calculator online. Ideal for pre-algebra, number theory, and homework like "find the prime factorization of 360."
- Trust and scopeCalculations run entirely in your browser; no data is sent. Trial division is deterministic and matches standard curriculum. Handles 1 with a clear explanation; input validated up to 12 digits.
This prime factorization calculator finds the prime factorization of any number using trial division. Enter a positive integer and get the exponential form (e.g. 100 = 2ยฒ ร 5ยฒ), a Factor List (2, 2, 5, 5), status tags (Prime, Composite, Even/Odd, Perfect square/cube), an interactive factor tree with terminal primes highlighted in green, and a Logic Trace of every division step plus a final step that collects terminal factors and converts to power notation. Copy results in one line:
n = p^a * q^b (p, p, q, q). Supports integers up to 12 digits; 1 is handled explicitly.Prime Factorization and the Fundamental Theorem of Arithmetic
By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization (up to the order of factors). Trial division finds it: divide by the smallest prime (2) while the number is even; when the quotient is odd, try 3, then 5, 7, and so on until the quotient is 1 or prime. Each divisor is a prime factor; the Logic Trace shows every step (e.g. "50 is even, divide by 2 โ 25"). The final step collects all terminal prime factors from the factor tree and writes them in exponential formโreinforcing the link between the tree and the formula.
The Factor Tree and How to Read Terminal Primes
The factor tree visualizes trial division: the root is n; the left child is the smallest prime factor, the right child is the quotient. Recursively repeat until every branch ends at a prime. Those leaf nodes are the terminal prime factorsโhighlighted in green in this calculator, with a "Primes" legend for clarity. Reading them left-to-right (or in tree order) gives the Factor List; grouping by value gives the exponential form. The tree is scrollable when large so nodes stay readable. Use it to teach how to find prime factors or to verify the prime factorization of 100 (2ยฒ ร 5ยฒ) and other common homework numbers.
FAQ
? What is prime factorization?
Prime factorization is writing a number as a product of primes raised to powers (e.g. 100 = 2ยฒ ร 5ยฒ). Every integer > 1 has a unique factorization (up to order). This calculator uses trial division and shows both the expanded Factor List (2, 2, 5, 5) and the exponential form.
? How do you find the prime factorization of a number?
Use trial division: divide by 2 while the number is even, then by 3, 5, 7, โฆ up to โn. Record each prime divisor; the quotient becomes the next number. The Logic Trace shows every step (e.g. "100 is even, divide by 2 โ 50"); the Factor Tree shows the same splits with the smallest prime on the left.
? What do the green and gray nodes mean in the Factor Tree?
Green nodes are prime factors (terminal nodes with no further factors). Gray nodes are composites that branch into a smallest prime (left) and the remaining quotient (right). The "Primes" legend under the tree labels the green nodes. Reading green nodes in order gives the Factor List.
? What is the difference between Factor List and exponential form?
The Factor List is the full sequence of primes (e.g. 2, 2, 5, 5). The exponential form groups repeats (e.g. 2ยฒ ร 5ยฒ). They are equivalent. The Logic Trace ends with "Collect all terminal prime factors" showing the product string then the conversion to power notation. Copy results includes both formats.
? What is the maximum number I can factor?
Positive integers from 2 to 12 digits (999,999,999,999). 1 has no prime factorization and is handled with a clear message. All calculations run in your browser; no data is sent to any server.