The integer n divides the integer m

iff there is an integer x such that nx = m. If you want to

be real fancy about it, you can write n | m (that's the

pipe, in case you couldn't tell).

Divisibility is symmetric since a | a for any integer a,

and it is transitive, since a | b and b | c imply that a | c.

Also, if a | b and a | c, then a divides any linear combination of the two: a | (bx + cy) for any integers x and y.

Note that b | 0 for any integer b, but we never, ever think of writing anything so unthinkable as 0 | b.

In fact, just writing b | c implies that b and c are both integers, b != 0

For positive b and c, b | c implies that b is less than or equal to c.

For a prime number p and positive b, b | p implies that b = 1 or b = p.

In the case that a does not divide b, there are distinct integers q and r such that a = bq + r, and 0 < r < b.

This is called the division algorithm. If you have a pocket calculator, you can calculate q and r quite simply:

the integer part of b / a is q, and a mod b is r.