Prime Factorization Calculator

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 = 2² × 5²), 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.

The calculator uses trial division, the most intuitive method for prime factorization. Starting with the smallest prime (2), it divides the input number repeatedly while it is divisible, records each factor, then moves to the next candidate (3, 5, 7, …) up to the square root of the remaining quotient. If the quotient is still greater than 1 after all candidates are exhausted, that quotient is itself prime. By the fundamental theorem of arithmetic, every integer greater than 1 has exactly one prime factorization (up to the order of factors). Worked example: To factor 360, divide by 2 three times (360 → 180 → 90 → 45), then by 3 twice (45 → 15 → 5), then by 5 once (5 → 1). The result is 2³ × 3² × 5, which the Factor List shows as 2, 2, 2, 3, 3, 5. The Logic Trace displays each division step, and the factor tree renders them visually with green terminal nodes for the prime factors.

Type any positive integer from 2 to 999,999,999,999 in the input field. The number 1 has no prime factorization and is handled with an explanatory message. Results appear instantly: the exponential form (e.g., 2³ × 3² × 5) on the result card, the expanded Factor List (2, 2, 2, 3, 3, 5) below it, and status tags indicating whether the number is prime or composite, even or odd, and whether it is a perfect square or cube. The interactive factor tree shows each split, left child is the smallest prime factor, right child is the remaining quotient, with green nodes marking terminal primes and gray nodes marking composites that split further. The Logic Trace walks through every division step. Use the copy button to export both exponential form and Factor List in one line for homework, notes, or further calculations.

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 → 25"). The final step collects all terminal prime factors from the factor tree and writes them in exponential form, reinforcing the link between the tree and the formula.

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 factors, highlighted 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 (2² × 5²) and other common homework numbers.