Decompose any integer into its unique prime factors with exponential form, expanded notation, and step-by-step division steps.
The Fundamental Theorem of Arithmetic (Unique Factorization Theorem) states that every integer greater than 1 either is a prime number itself or can be represented uniquely as the product of prime numbers, up to the order of the factors. Prime factorization is the mathematical foundation of number theory, cryptography (such as RSA public-key encryption), and algorithmic hash functions.
Divide repeatedly by the smallest prime 2 until the quotient is odd, then test 3, 5, 7, and successive odd numbers up to the square root √n.
Repeatedly branch composite numbers into smaller factors until every terminal leaf node is an indivisible prime number.
Two integers are coprime if their prime factorizations share no identical prime bases in common.
The total number of divisors of n equals (a₁ + 1)(a₂ + 1)...(aₖ + 1).
Find the prime factorization and canonical exponential form of 360.
If 1 were considered prime, the Fundamental Theorem of Arithmetic would fail to be unique, because any number could be written with arbitrarily many factors of 1 (e.g. 6 = 2 × 3 = 1 × 2 × 3 = 1 × 1 × 2 × 3).
RSA public-key cryptography relies on the mathematical asymmetry that multiplying two large primes together (e.g. 1024-bit primes) is instantaneous, but factoring their product back into primes is virtually impossible for modern supercomputers without knowing the private key.
The largest known primes are Mersenne primes of the form 2ᵖ - 1, containing tens of millions of digits, discovered by the Great Internet Mersenne Prime Search (GIMPS).
This is the Goldbach Conjecture (formulated in 1742), one of the oldest and most famous unsolved problems in mathematics.
Understanding your calculation
Prime factorization expresses a positive integer greater than one as a product of primes, grouping repeated primes as powers.
360 = 2³ × 3² × 5. Multiplying those factors gives the original integer.
One is not prime and has no prime factorization. Zero and noninteger values are outside ordinary positive-integer prime factorization.