Tag Info

This tag is for questions about divisibility, that is, determining when one thing is a multiple of another thing.

If $$a$$ and $$b$$ are integers, $$a$$ divides $$b$$ if $$b=ca$$ for some integer $$c$$. This is denoted $$a\mid b$$. It is usually studied in introductory courses in number theory, so add if appropriate.

A common notation used for the phrase "$$a$$ divides $$b$$" is $$a|b$$. It is also common to negate the notation by adding a slash like this: "$$c$$ does not divide $$d$$" written as $$c\nmid d$$. Note that the order is important: for example, $$2|4$$ but "$$4\nmid 2$$".

This notion can be generalized to any ring. The definition is the same: For two elements $$a$$ and $$b$$ of a commutative ring $$R$$, $$a$$ divides $$b$$ if $$ac=b$$ for some $$c$$ in $$R$$.

Divisibility in commutative rings corresponds exactly to containment the poset of principal ideals. That is, $$a$$ divides $$b$$ if and only if $$aR\subseteq bR$$. For commutative rings like principal ideal rings, this means that divisibility mirrors exactly the poset of all ideals of the ring.

The topics appropriate for this tag include, for example:

• Questions about the relation $$\mid$$.
• Questions about the GCD and LCM.

There are divisibility rule that is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits.