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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.

Ways to test for divisibility by single-digit numbers in base 10:

1
This is here for completeness. Every integer is divisible by 1.
2
The last digit is 0, 2, 4, 6, or 8.
3
The sum of the digits is divisible by 3. (If you are dealing with a freaking huge number you can apply this recursively.)
4
The last two digits form a number which is itself divisible by 4.
5
The last digit is 0 or 5.
6
The sum of the digits is divisible by 3 and the last digit is 0, 2, 4, 6, or 8.
7
I defer to how to determine whether a number is divisible by 7.
8
Either:
• The hundreds place is even, and the last two digits form a number which is divisible by 8.
• orThe hundreds place is odd, and the last two digits minus 4 form a number which is divisible by 8.
9
The sum of the digits is divisible by 9.

Divisibility is not symmetric. a | b usually doesn't imply b | a. The property that a | a means that divisibility is reflexive.
To find out if an integer is divisible by 11:

Consider the integer a where

a = an10n + an-110n-1 +...+ a1101 + a0

11 divides a if and only if the alternating sum of the digits (-1)nan + (-1)n-1an-1 + ... - a1 + a0 is divisible by 11.

eg. Is 152328 divisible by 11?
Observe -1 + 5 - 2 + 3 - 2 + 8 = 11
11 obviously divides 11, therefore 152328 is divisible by 11.

Di*vis`i*bil"i*ty (?), n. [Cf. F. divisibilit'e.]

The quality of being divisible; the property of bodies by which their parts are capable of separation.

Divisibility . . . is a primary attribute of matter. Sir W. Hamilton.

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