Divisibility

In elementary mathematics, divisibility is a relation between two natural numbers: a number d divides a number n, if n is the product of d and another natural number k. Since this is a very common notion there are many equivalent expressions: d divides n wholly or evenly (if one wants to put emphasis on it), d is a divisor or factor of n, n is divisible by d, or (conversely) n is a multiple of d.

Every natural number n has two divisors, 1 and n, which therefore are called trivial divisors. Any other divisor is called a proper divisor. A natural number (except 1) which has no proper divisor is called prime, a number with proper divisors is called composite.

The concept of divisibility can obviously be extended to the integers. In the integers, every integer n has four trivial divisors: 1, -1, n, -n. Because of 0 = 0.n, any n is divisor of 0, and 0 divides only 0.

Further extensions include algebraic integers, polynom rings. and rings in general. (However, divisibility is useless for rational or real numbers: Because of ad = b for d=b/a every rational or real number divides every other rational or real number.)

Some properties:
 * 1) a is divisor of a,
 * 2) if a is a divisor of b, and b is a divisor of a, then a equals b,
 * 3) if a is divides b, and b divides c, then a divides c,
 * 4) if a divides b and c then it also divides a+b.

The following property is important and frequently used in number theory. Therefore it is also called Fundamental theorem of number theory.
 * If a prime number divides a product ab, and it does not divide a, then it divides b.

Properties (1-3) show that "is divisor of" can be seen as a partial [[order relation|order}} on the natural numbers. In this order,
 * 1 is the minimal element since it divides all numbers, and
 * 0 is the maximal element since it is a multiple of every number,
 * the greatest common divisor is the greatest lower bound (or infimum), and
 * the least common multiple is the smallest upper bound (or supremum).

In mathematical notation, "a divides b" is written as
 * $$ a\mid b $$

Using this notation, and
 * $$ a,b,c,d,k,n,p \in \mathbb N, p \ \textrm{prime} $$

the definition of is divisor of is
 * $$ d \mid n :\Leftrightarrow (\exist k) dk = n $$

and the properties are The Fundamental Theorem is and the definition of the order &mdash; if one wants to avoid the vertical bar &mdash; is given by
 * 1) $$ a \mid a $$
 * 2) $$ a \mid b \;,\ b \mid a \Rightarrow a=a $$
 * 3) $$ a \mid b \;,\ b \mid c \Rightarrow a \mid c $$
 * 4) $$ a \mid b \;,\ a \mid c \Rightarrow a \mid (b+c) $$
 * $$ p \mid ab \;,\ p \not\mid a \Rightarrow p \mid b $$
 * $$ a \le b :\Leftrightarrow a \mid b $$
 * $$ a \le b :\Leftrightarrow a \mid b $$