Divisors

Since multiplication is not commutative, there are two versions of the Euclidean algorithm, one for right divisors and one for left divisors.

What is the probability that it will have many more or many fewer divisors or prime divisors than the average?

All elements of this form are zero divisors (also see the section "Division").

All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction contain zero divisors.

Any positive divisor of is a product of prime divisors of raised to some power.

A semiperfect number is a natural number that is equal to the sum of all or some of its proper divisors.

A semiperfect number that is equal to the sum of all its proper divisors is a perfect number.

Examples Plot of the number of divisors of integers from 1 to 1000.

For example, the computation of polynomial greatest common divisors is systematically used for the simplification of expressions involving fractions.

Further: how many prime divisors will it have, on average?

However, the ring constructed in this way contains zero divisors and thus cannot be a field.

If H is a subgroup of this group, its order (the number of elements) must be a divisor of the order of G which is 3. Since 3 is prime, its only divisors are 1 and 3, so either H is G, or H is the trivial group.

If x and y are elements of a PID without common divisors, then every element of the PID can be written in the form ax + by.

Indeed, they are (trivially) zero divisors and clearly form an ideal of the associative algebra (and thus ring ) of the dual numbers.

In particular, because the prime numbers are the numbers with no nontrivial divisors, a cyclic group is simple if and only if its order (the number of its elements) is prime.

Knowledge of the divisors of the group order G can often be used to gain information about the order of the center or of the conjugacy classes.

Long division is used for larger integer divisors.

One inefficient approach to finding the GCD of two natural numbers a and b is to calculate all their common divisors; the GCD is then the largest common divisor.

Prime numbers have exactly 2 divisors, and highly composite numbers are in bold.

Taking instead each number to the set of its prime power divisors defines a map g: ℕ → ℙ(ℕ) that is order-preserving, order-reflecting, and hence an order-embedding.