Divisibility

In elementary mathematics, divisibility is a relation between two natural numbers: a number d  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 or  (if one wants to put emphasis on it), d is a divisor or  of n, n is  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 generalizations are to 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 a 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 divides b, and b divides c, then a divides c,
 * 4) if a divides b and c then it also divides b+c (or, more generally, kb+lc for arbitrary integers k and l).

The following property is important and frequently used in number theory. Therefore it is also called Fundamental lemma 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 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
 * $$ a,b,c,d,k,l,n,p \in \mathbb N, p \ \textrm{prime} $$

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

and the properties listed are written as
 * 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) $$ a \mid b \;,\ a \mid c \Rightarrow a \mid (kb+lc) $$

The Fundamental Lemma is and the definition of the order &mdash; if one wants to avoid the vertical bar &mdash; is given by
 * $$ p \mid ab \;,\ p \not\;\mid a \Rightarrow p \mid b $$
 * $$ a \le b :\Leftrightarrow a \mid b $$