Divisor
Further notions and facts

There are some elementary rules:

If a ∣ b c, {\displaystyle a\mid bc,} and gcd (a, b) = 1, {\displaystyle \gcd(a,b)=1,} then a ∣ c. {\displaystyle a\mid c.} This is called Euclid's lemma.

If p {\displaystyle p} is a prime number and p ∣ a b {\displaystyle p\mid ab} then p ∣ a {\displaystyle p\mid a} or p ∣ b. {\displaystyle p\mid b.}

A positive divisor of n {\displaystyle n} that is different from n {\displaystyle n} is called a proper divisor or an aliquot part of n {\displaystyle n} (for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly divide n {\displaystyle n} but leaves a remainder is sometimes called an aliquant part of n. {\displaystyle n.}

An integer n > 1 {\displaystyle n>1} whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer that has exactly two positive factors: 1 and itself.

Any positive divisor of n {\displaystyle n} is a product of prime divisors of n {\displaystyle n} raised to some power. This is a consequence of the fundamental theorem of arithmetic.

A number n {\displaystyle n} is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisors is less than n, {\displaystyle n,} and abundant if this sum exceeds n. {\displaystyle n.}

The total number of positive divisors of n {\displaystyle n} is a multiplicative function d (n), {\displaystyle d(n),} meaning that when two numbers m {\displaystyle m} and n {\displaystyle n} are relatively prime, then d (m n) = d (m) × d (n). {\displaystyle d(mn)=d(m)\times d(n).} For instance, d (42) = 8 = 2 × 2 × 2 = d (2) × d (3) × d (7) {\displaystyle d(42)=8=2\times 2\times 2=d(2)\times d(3)\times d(7)}; the eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However, the number of positive divisors is not a totally multiplicative function: if the two numbers m {\displaystyle m} and n {\displaystyle n} share a common divisor, then it might not be true that d (m n) = d (m) × d (n). {\displaystyle d(mn)=d(m)\times d(n).} The sum of the positive divisors of n {\displaystyle n} is another multiplicative function σ (n) {\displaystyle \sigma (n)} (for example, σ (42) = 96 = 3 × 4 × 8 = σ (2) × σ (3) × σ (7) = 1 + 2 + 3 + 6 + 7 + 14 + 21 + 42 {\displaystyle \sigma (42)=96=3\times 4\times 8=\sigma (2)\times \sigma (3)\times \sigma (7)=1+2+3+6+7+14+21+42}). Both of these functions are examples of divisor functions.

If the prime factorization of n {\displaystyle n} is given by

then the number of positive divisors of n {\displaystyle n} is

and each of the divisors has the form

where 0 ≤ μ i ≤ ν i {\displaystyle 0\leq \mu _{i}\leq \nu _{i}} for each 1 ≤ i ≤ k. {\displaystyle 1\leq i\leq k.}

For every natural n, {\displaystyle n,} d (n) < 2 n. {\displaystyle d(n)<2{\sqrt {n}}.}

Also,

where γ {\displaystyle \gamma } is Euler–Mascheroni constant.One interpretation of this result is that a randomly chosen positive integer n has an averagenumber of divisors of about ln n. {\displaystyle \ln n.} However, this is a result from the contributions of numbers with "abnormally many" divisors.
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