# Category:Divisors

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This category contains results about **Divisors** in the context of **Algebra**.

Definitions specific to this category can be found in Definitions/Divisors.

Let $\struct {\Z, +, \times}$ be the ring of integers.

Let $x, y \in \Z$.

Then **$x$ divides $y$** is defined as:

- $x \divides y \iff \exists t \in \Z: y = t \times x$

## Subcategories

This category has the following 23 subcategories, out of 23 total.

### A

- Aliquot Sums (2 P)

### C

### D

- Division Theorem (24 P)
- Divisor Divides Multiple (3 P)

### E

### F

- Feit-Thompson Conjecture (3 P)

### G

### I

### P

- Primitive Prime Factors (1 P)
- Product of Divisors (7 P)

## Pages in category "Divisors"

The following 69 pages are in this category, out of 69 total.

### A

### C

### D

- Divides is Reflexive
- Divisibility of Elements of Pythagorean Triple by 7
- Divisibility of Fibonacci Number
- Divisibility of Fibonacci Number/Corollary
- Divisibility of Product of Consecutive Integers
- Division Theorem
- Divisor Divides Multiple
- Divisor is Reciprocal of Divisor of Integer
- Divisor of Product
- Divisor of Sum of Coprime Integers
- Divisor Relation is Antisymmetric
- Divisor Relation is Transitive
- Divisor Relation on Positive Integers is Partial Ordering
- Divisor Relation on Positive Integers is Well-Founded Ordering
- Divisors obey Distributive Law
- Divisors of Factorial
- Divisors of Negative Values
- Divisors of One

### I

- Integer Divided by Divisor is Integer
- Integer Divides its Absolute Value
- Integer Divides its Negative
- Integer Divides Itself
- Integer Divides Zero
- Integer Divisor Results
- Integer Divisor Results/Divisors of Negative Values
- Integer Divisor Results/Integer Divides its Absolute Value
- Integer Divisor Results/Integer Divides its Negative
- Integer Divisor Results/Integer Divides Itself
- Integer Divisor Results/Integer Divides Zero
- Integer Divisor Results/One Divides all Integers

### N

- Natural Number is Divisor or Multiple of Divisor of Another
- Natural Numbers with Divisor Operation is Isomorphic to Subgroups of Integer Multiples under Inclusion
- Natural Numbers with Divisor Operation is Isomorphic to Subgroups of Integer Multiples under Inclusion/Corollary
- Non-Zero Integer has Finite Number of Divisors
- Number divides Number iff Cube divides Cube
- Number divides Number iff Square divides Square
- Number does not divide Number iff Cube does not divide Cube
- Number does not divide Number iff Square does not divide Square

### O

### P

### S

- Sequence of Numbers Divisible by Sequence of Primes
- Set of 3 Integers each Divisor of Sum of Other Two
- Set of Common Divisors of Integers is not Empty
- Smallest Integer Divisible by All Numbers from 1 to 100
- Square Divides Product of Multiples
- Subtraction of Multiples of Divisors obeys Distributive Law
- Sum of Squares of Divisors of 24 and 26 are Equal
- Sum Over Divisors Equals Sum Over Quotients
- Summation of Summation over Divisors of Function of Two Variables