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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.
ຕົວຢ່າງທີ່ໄດ້ເຮັດ · divisors of 36
ຂັ້ນຕອນຕໍ່ຂັ້ນຕອນ
- 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.
ເປີດເຜີຍຄຳຕອບ
ປັດຈຸບັນທ່ານ ເລືອກບັນຫາ ຫຼື ພິມ ຫຼື ແຕ້ມເອງ. ທຸກບາດກ້າວ, ຮູບ, ຄໍາຕອບຖືກລົບຈົນກວ່າເຈົ້າຈະຖາມມັນ.
ຕົວສະແດງທີ່ໃຊ້ຢູ່ທີ່ນີ້
ກົດຕົວອັກສອນໃດໜຶ່ງເພື່ອເບິ່ງຄວາມໝາຍເຕັມ, ຮູບ ແລະ ຕົວອັກສອນແຕ່ລະຕົວໃນມັນໝາຍຄວາມວ່າແນວໃດ.
ວິທີການ: 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.
ຄໍາຖາມທີ່ຄົນຖາມ
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.
ເພີ່ມເຕີມໃນ Number Theory
Prime factorisationPrime numbersGCD and LCMModular arithmeticSequencesNumber basesDiophantine equationsFermat's little theorem and Euler's theoremRSA: cryptography from number theory