Main Index
Number Theory
Arithmetics
Multiplication
Divisibility
Subject Index
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Main Index
Number Theory
Arithmetics
Multiplication
Divisibility
Subject Index
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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:
and 
, then 
, that is divisibility relation is transitive,
and 
, then 
or 
,
and 
, then 
for all integers 
and 
.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.