360
- Input
- 360
- Expected output
- 2³ × 3² × 5
Factors 2, 3 and 5 with exponents 3, 2 and 1: adding one to each exponent and multiplying, (3+1)×(2+1)×(1+1) = 24, gives the total number of divisors of 360.
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prime factorization of integers
By the Fundamental Theorem of Arithmetic, every integer greater than 1 has a unique prime factorization, aside from order: 360 = 2³ × 3² × 5. The tool tests divisors only up to the square root of whatever remains at each step, which is enough to prove no larger hidden factor exists.
Factors 2, 3 and 5 with exponents 3, 2 and 1: adding one to each exponent and multiplying, (3+1)×(2+1)×(1+1) = 24, gives the total number of divisors of 360.
All three exponents (2, 2 and 2) are even, which is enough to prove 1764 is a perfect square: 42² = 1764.
840 is divisible by every integer from 1 to 8 because its factorization gathers 2³, 3, 5 and 7, enough factors to cover each of them.
A prime number is an integer greater than 1 that has no positive integer divisors other than 1 and itself. The first primes are 2, 3, 5, 7, 11, 13…
If every exponent in the factorization is even, the number is a perfect square: 36 = 2² × 3² has both exponents even, so it is (2×3)² = 6². Meanwhile 12 = 2² × 3 has one odd exponent (the one on 3), so it is not.
Up to 10²⁴, a 25-digit number, the practical limit of the BigInt it uses internally. Trial division only needs to test divisors up to the square root of the remaining value, but near the ceiling that can still mean up to 10¹² candidates in the worst case.
No. Checking whether a number is prime uses Miller-Rabin with 12 fixed witnesses, correct for any value below 3.3 × 10²⁴, and answers yes or no quickly. Factoring requires finding every factor, a more expensive problem that the tool solves through trial division.
Supports positive integers up to 10²⁴.
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