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Divisors
Counting and listing divisors from the prime factorisation.
If n = p₁ᵃ · p₂ᵇ · …, then every divisor picks an exponent between 0 and a for p₁, between 0 and b for p₂, and so on, which is why the number of divisors is (a+1)(b+1)…. Perfect squares are exactly the numbers with an odd count.
Voorbeeld van werk · divisors of 36
Stap voor stap
- 36 = 2^{2} \times 3^{2}
Prime-factorise first.
- d(n) = (2+1) \cdot (2+1) = 9
Each divisor picks an exponent from 0 up to the prime's exponent, so multiply (exponent + 1) for each prime.
- 1, 2, 3, 4, 6, 9, 12, 18, 36
List them in increasing order.
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Hoe: Divisors
- Prime-factorise first.
- Each divisor picks an exponent from 0 up to the prime's exponent, so multiply (exponent + 1) for each prime.
- List them in increasing order.
Vragen die mensen stellen
Why are primes so important?
Every integer factors into primes in exactly one way, so primes are the atoms of multiplication. Cryptography relies on that factoring being easy to state and hard to do.
How do I tell whether a big number is prime?
Trial division up to the square root works for small numbers. For large ones, probabilistic tests (Miller-Rabin) give an answer that is wrong with negligible probability, and deterministic tests (AKS) exist but are slower.
Meer in Number Theory
Prime factorisationPrime numbersGCD and LCMModular arithmeticSequencesNumber basesDiophantine equationsFermat's little theorem and Euler's theoremRSA: cryptography from number theory