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divisor

In mathematics, a divisor of an integer n is an integer d such that n = d·q for some integer q. When working with positive divisors, d is taken to be a positive integer. In that sense, d is a divisor of n if and only if n mod d = 0. Every positive integer n has at least two positive divisors: 1 and n. Negative divisors exist as well, since if d is a divisor, so is -d, but positive divisors are typically the focus in elementary contexts.

Examples help illustrate the concept. The positive divisors of 28 are 1, 2, 4, 7, 14, and

Divisors are closely tied to prime factorization. If n has prime factorization n = p1^a1 p2^a2 ... pk^ak,

28.
Proper
divisors
are
the
divisors
other
than
the
number
itself,
so
for
28
the
proper
divisors
are
1,
2,
4,
7,
and
14.
The
notion
extends
to
other
integers
in
the
same
way,
with
the
set
of
divisors
reflecting
the
number’s
factorization.
then
the
number
of
positive
divisors
of
n
is
tau(n)
=
(a1+1)(a2+1)...(ak+1).
Divisors
also
come
in
pairs:
for
every
divisor
d,
n/d
is
another
divisor,
and
d
=
sqrt(n)
when
n
is
a
square.
Divisors
underpin
many
arithmetic
concepts,
including
the
greatest
common
divisor,
least
common
multiple,
and
tests
related
to
primality
and
perfect
numbers.