When you divide an integer by another, and if you are not interested in fractions, you will get a quotient and remainder. You know this already. Let us make this more precise. The usual mathematical way to express the idea of division with remainder is not to talk about the division itself, but about reconstructing the dividend from the quotient, divisor, and remainder; the idea is simple and in the air:
Check that the quotients and remainders in the examples satisfy the criteria.
The mathematical formulation of the above ideas can be proved, and is sometimes called the division theorem.
Theorem: Let D and d be integers, and d be non-zero. (We are dividing D by d.) Then for some integers quotient and remainder, we have
D = quotient * d + remainder, and
0 <= remainder < |d|;
furthermore, quotient and remainder are unique: if
D = q * d + r, and
0 <= r < |d|,
then q = quotient and r = remainder.
Proof: Deal with existence first. For natural number D, use induction. Then extend the existence to negative D. Finally, deal with the uniqueness part, noting that remainder - r must be 0.
Again, check that the examples satisfy this theorem.
If you are only interested in the natural numbers, then you do not need to worry about negative numbers, so you can drop the absolute signs above.
If the remainder happens to be 0, i.e., for some integer quotient, we have
D = quotient * d
then we can say the followings:
For example, 11 is not a multiple of 4, and 4 is not a divisor of 11; 12 is a multiple of 4, 4 is a divisor of 12, and 4 divides 12.
For some reasons I do not like to use the word ``factor'' in this way, but it is just a preference.
These words and concepts play a next-to-central role in number theory; therefore I take this opportunity to introduce them.