text: # Definition:Divisor (Algebra)/Integer
  
  ## Definition
  
  Let $\struct {\Z, +, \times}$ be the ring of integers.
  
  Let $x, y \in \Z$.
  
  Then $x$ divides $y$ is defined as:
  
  $x \divides y \iff \exists t \in \Z: y = t \times x$
  
  ### Aliquot Part
  
  An aliquot part of an integer $n$ is a divisor of $n$ which is strictly less than $n$.
  
  ### Aliquant Part
  
  An aliquant part of an integer $n$ is a positive integer which is less than $n$ but is not a divisor of $n$.
  
  ## Terminology
  
  Let $x \divides y$ denote that $x$ divides $y$.
  
  Then the following terminology can be used:
  
  $x$ is a divisor of $y$
  $y$ is a multiple of $x$
  $y$ is divisible by $x$.
  
  In the field of Euclidean geometry, in particular:
  
  $x$ measures $y$.
  
  To indicate that $x$ does not divide $y$, we write $x \nmid y$.
  
  ## Notation
  
  The conventional notation for $x$ is a divisor of $y$ is "$x \mid y$", but there is a growing trend to follow the notation "$x \divides y$", as espoused by Knuth etc.
  
  The notation '$m \mid n$' is actually much more common than '$m \divides n$' in current mathematics literature. But vertical lines are overused -- for absolute values, set delimiters, conditional probabilities, etc. -- and backward slashes are underused. Moreover, '$m \divides n$' gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.
  
  An unfortunate unwelcome side-effect of this notational convention is that to indicate non-divisibility, the conventional technique of implementing $/$ through the notation looks awkward with $\divides$, so $\not \! \backslash$ is eschewed in favour of $\nmid$.
  
  Some sources use $\ \vert \mkern -10mu {\raise 3pt -} \$ or similar to denote non-divisibility.
  
  ## Also known as
  
  A divisor can also be referred to as a factor.
  
  ## Examples
  
  ### $2$ divides $4$
  
  $2 \divides 4$
  
  ### $3$ divides $12$
  
  $3 \divides 12$
  
  ### $4$ divides $\paren {-12}$
  
  $4 \divides \paren {-12}$
  
  ### $2$ does not divide $5$
  
  $2 \nmid 5$
  
  ### $3$ does not divide $4$
  
  $3 \nmid 4$
  
  ### $3$ does not divide $10$
  
  $3 \nmid 10$
  
  ## Also see
  
  • Results about divisors can be found here.
  
  ## Historical Note
  
  The concept of divisibility has been studied for over $3000$ years.
  
  from before the time of Pythagoras of Samos, the ancient Greeks were considering questions about:
  
  odd and even numbers.
  perfect numbers
  amicable numbers
  prime numbers
  
  as well as many other.
  
  Even today, some of those questions remain unanswered.