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Divisibility is the relation between numbers. Let and be numbers. We say that divides if there exists such a number such that:

or, divides if *is a multiple of* .

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The above equality can also be transformed into a more familiar form:

where

is called

*divisor*is called

*dividend*is called

*quotient*.

The divisibility relation between numbers is denoted by:

and read as: *b divides a*.

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There are infinites of multiples of any number, and there are finally many divisors.

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We say that a relation is reflexive because each number is considered to divide itself.

If , then .

We say that a relation is antisymmetric because it holds that if two numbers divide each other, then the numbers are the same.

If , and , then .

We say that a relation is transitive because it holds that when a number divides the number and that same number divides the number , then the number also divides the number .

If , and , then .

The definition of divisibility can be extended to integers. Let and be integers. The number divides the number if there exists an integer that is valid.

It is also true that 0 is divisible by any non-zero integer , since

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Each integer has *at least* 4 divisors:

Let and be natural numbers. Then, their:

sum:

product:

are natural numbers, but the quotient is not necessarily a natural number. When the quotient is a natural number, we know that the numbers and are in the divisibility relation, otherwise they are not in the relation and do not return a natural number to us when dividing.

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Regardless of the fact whether two natural numbers are in the divisibility relation or not, for all pairs of natural numbers , where , we can write the basic theorem on division.

The basic theorem on division says that the division of any two natural numbers and , to which and apply, can be written as

where:

divisor

dividend

quotient

remainder where .

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material editor: ISMAIL SHOBOLA