0 is always a special case

The standard definition of divisibility (in number theory) is

Definition. If a and b are integers and there is some integer c such that a = b · c, then we say that b divides a or is a factor or divisor of a and write b|a.

So 3 trivially divides 0 (as do all integers).

If one were to define "multiple" I can't think of any reason to do so by using divisibility (you can readily modify the above definition to show that multiples can be defined by multiplying), but if you wanted to there is no reason to exclude 0 from the numerator. Remember the integers aren't a field (i.e. 0|0 is totally legit).

By using the notation "Z" you have said a/b is an element of all integers. Firstly, a/b isn't an integer it is a quotient, Q. Secondly, Integers all {..., -2, -1, 0, 1, 2,...} so the definition you've used is flawed.

I strongly, vehemently agree with this, from another angle. For an integer (x) to be a multiple of another integer (y), you must be able to divide the integer (x) by some integer (z) to result in (y). There is nothing you can divide into zero to equal 3. Nothing at all. Therefore 0 is not only not a multiple of three, it's not a multiple of anything. Zero is not a multiple of anything because anything multiplied by zero equals zero. And there is no other possible solution.

A múltiple of a number gives as result an int number when you put the "multiple" over that number as you said but look... eg. 3/0 I think it's undefined because that is an infinite :)

You can't even spell 'integer' mate... Zero is in the timestable of every integer.

in group number theory the number 0 is included in the times tables and in the numbers defined as Z

Everything has a multiple of 0

0x0 0x1 0x2

etc.

0x3 = 0 1x3 = 3 2x3 = 6

Yes. 0 is a multiple of any number. You can always have zero of somethig..

Lemons come in bags of 5 in Australia... So if I go shopping I can buy 1 bag which = 5 lemons 2 bags which = 10 lemons

Or I can decide I don't want lemons. And buy 0 bags of 5 lemons...which = 0

I seem to be really fixated on fruit shopping :p

Zero doesn't go into 3 at all because by definition you would have to divide 3 by 0, which is undefined.

You are wrong. You're defining factors, not multiples.

in math you cannot use infinity in a multiplication like that as if it's a "number"...

If you multiplied infinity times zero you would still get 0. Zero is zero and there is no number that you could multiply it by to get anything other than zero. Therefore zero is not a multiple of three. Zero is merely a place holder or representation for nothing (or null in my world).

That is just a way of explaining the notion of multiple.If $a=bc$ then $a$ is called a multiple of $b$ and $c$ .So, for $0$ to be a mulitple of $3$ , $0$ must equal to $3b$ ,for some $b$ .Hence $0=3b\rightarrow b=\frac{0}{3}=0$ .Hence $0=3\times 0$ so 0 is a multiple of 3.Real life analogies sometimes fail.Let me give you an example:-3 is a multiple of 3,albeit a negative multiple,but a multiple nonetheless.But how can we put things into groups of $3$ (-1) times in real life?We can't.

You've confused factors and multiples. 3 has only two factors, i.e., it itself is a multiple of only two numbers, 1 and 3. But, 3 has an infinite number of multiples, like 0,3,6,9 and so on.

That's a little convoluted way of thinking about it.. but if you take your example and apply it to the definition of a multiple: "b is a multiple of a if b = na for some integer n" Then you have: one "Cake" is a multiple of "Parts" for some integer 3.

Therefore, zero "Cake" is a multiple of "Parts" for some integer 0. ie: no pieces equal no cake.

Or if you had no cake and divided it equally among three people, they would each have no pieces.

If you divide 0 cakes into 3 parts then each part has size 0, so 0 is indeed a multiple of 3, and 3 in that sense a divisor of 0. (But in the way mathematicians usually construe the phrase "divisor of zero" it means a nontrivial divisor of zero: you multiply two nonzero things and get zero. This is impossible with numbers but possible with matrices and with congruence classes in modular arithmetic when the modulus is a composite number.)

Whoops I was thinking about factors, not multiples.

Pamela, can you find a source that says that n has to be a natural number? The common definition states that n has to be an integer.