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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.

Ohatra · divisors of 36

Number-Theory · Divisors

Divisors of 36

36

Dingana amin'ny dingana

  1. 36 = 2^{2} \times 3^{2}

    Prime-factorise first.

  2. 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.

  3. 1, 2, 3, 4, 6, 9, 12, 18, 36

    List them in increasing order.

Asehoy ny valinteny
1, 2, 3, 4, 6, 9, 12, 18, 36

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Ahoana ny fomba: Divisors

  1. Prime-factorise first.
  2. Each divisor picks an exponent from 0 up to the prime's exponent, so multiply (exponent + 1) for each prime.
  3. List them in increasing order.

Fanontaniana napetrak'ireo olona

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.

Mbola maro ao Number Theory

Prime factorisationPrime numbersGCD and LCMModular arithmeticSequencesNumber basesDiophantine equationsFermat's little theorem and Euler's theoremRSA: cryptography from number theory