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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.

דוגמה עובדת: divisors of 36

Number-Theory · Divisors

Divisors of 36

36

צעד אחר צעד

  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.

גלה את התשובה
1, 2, 3, 4, 6, 9, 12, 18, 36

Symbols used here

\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
a \mid b,\ \gcd(a,b)
divides, greatest common divisor
b is a multiple of a; the largest number dividing both.
\varphi(n),\ \pi(x)
Euler's totient, prime-counting function
Count of 1..n coprime to n; number of primes up to x.
\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n
integers modulo n
The remainders 0…n−1 with clock arithmetic.
a \bmod n
remainder
What is left after dividing a by n.

How to: 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.

Questions people ask

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.

נסה את שלך.

יותר בפנים. Number Theory

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