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Number Theory
Arithmetics
Multiplication
Divisibility
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An integer is called a divisor of an integer
, if
evenly divides
without leaving a remainder, that is if there is an integer
such that
. Divisor of
is also called a factor of
.
A positive divisor of which is different from
is called a proper divisor or an aliquot part of
. If a number does not evenly divide
but leaves a remainder it is called an aliquant part of
.
To generate the set of divisor of a positive integer go to .
If is a divisor of
we also say that
is divisible by
, or that
is a multiple of
.
Numbers and
divide (or are divisors of) every integer, and every integer (and its negative) is a divisor of itself. Therefore the divisors
of an integer
are called its trivial divisors. A divisor of
that is not
, or
is called a non-trivial divisor. Numbers having al least one non-trivial divisor are called composite numbers. In the opposite case they are called prime numbers.
The divisibility relation “ is a divisor of
” is a binary relation denoted by
. If
does not divide
we write
.
The basic properties of the divisibility relation are:
The relation of divisibility turns the set of non-negative integers into a partially ordered set. Moreover it is a a complete distributive lattice when the meet operation is given by the greatest common divisor and the join operation by the least common multiple. The largest element of this lattice is 0 and the smallest is 1.
Cite this web-page as:
Štefan Porubský: Divisors.