So we have integer numbers, positive and negative.

And division is not always possible over integers.

So we have to identify the cases when it is possible.

And for this, we have to understand first what we mean by division.

What does it mean that a is divisible by b?

Here's the naive answer we can come up with.

So just consider a rational number a/b,

if it is integer, then a is divisible by b.

If it is not integer,

then is not divisible by b.

And this is a correct statement,

but it is not very good as definition.

And the reason is that this definition reduces our question

to a notion that is more complex than the notions we are currently have.

The notion of rational numbers.

So to come up with better definition,

let's unwrap what we are currently trying to see.

What we are saying when we say that a/b is integer?

Okay. So, what does it mean that a/b is integer?

It means that the denominator cancels out in this fraction.

And it means that a can be represented as a product of two integers.

One of which is b,

and another is some other integer k. And this means that

a = b * k. And then we have that our fraction is b * k / b,

and b cancels out,

and the result is k. So,

a / b, is integer number.

So, a / b integer if a is representable as a product of two integers,

b and some other integer.

And now, this formulation uses only simple notions.

The notion of product and that's it.

And so, this will be our formal definition,

a is divisible by b.

Or b / a.

And we denote it like this,

if there is an integer k, such that a is the product of b,

and k. And if a is not divisible by b,

if b doesn't divide a,

we denote this in this way.

And this definition has a very simple intuitive sense.

So here is a definition of a divisibility again and the intuition is the following.

And for simplicity, assumed that a and b are positive.

So, suppose we have a objects,

and I would like to split them in groups of size b.

And then, it is possible if a is divisible by b,

and k will be the size of the number of groups.

So, we have k groups of the objects.

The number of objects in all groups is k * b,

and it should be equal to a.

Let's consider several examples.

For example, number 15 is divisible by three.

And indeed, we can pick k = 5,

and 15 = 3 * 5.

So 15 is divisible by three.

We have introduced our notion of divisibility for negative numbers also.

So for example, 12 is divisible by -4.

And here, we can pick k = -3.

So, 12 = -4 * -3,

and 12 is divisible by -4.

And one more example,

-24 is divisible by -6, and indeed here,

k can be equal to 4,

and -24 = -6 * 4.

And final example, 15 is not divisible by four.

And indeed, there is no such k,

such that 15 = 4 * k. There is no such integer k. So,

15 is not divisible by 4.

Okay. But why do we care about formal definition if everything is very simple?

So, note that in our examples are very simple.

So we are dealing with simple notions now.

Why do we care about some specific formal definition?

And one of the reasons is that,

once we have a formal definition,

we can show general purposes.

We can establish a general purposes of our notion.

And here is an example.

So, suppose number c divides a,

and the same number c,

divides b as well.

Then we can show that c divides sum of a, and b,

and difference of a and b as well.

And why this is true?

Note that since c divides a, and c divides b,

then we know that a and b has the following form.

a = c * k₁ for some k₁ and b = c * k₂ for some k₂.

Okay. And then, we can add or subtract a, and b.

And then obtain that.

a + b = c * k₁ + k₂, and a - b = c * k₁ - k₂,

and so, a +,

or a - b has the same form.

And by definition, a + b and a - b are also divisible by c. So,

using our formal definition,

we can establish with general property which is true for all numbers.

Here's another example, suppose b divides a.

Is it true that b divides 3 * a?

Yes, it turns out to be true.

And again, we can just use definition of divisibility.

b divides a, means that,

a is equal to b * k,

for some integer k. And now we can consider 3 * a.

And note that using this previous equation we see that b,

3 * a = b * 3k.

So, 3 * a again,

satisfies the definition of divisibility by b.

And so, b divides 3 * a.

And in the general case,

we have the following,

if b divides a,

and we have some integer c,

then b / a * c. And the definition of a proof here is basically the same.

Again, we can just look at the definition of divisibility.

b divides a, means that there is some k,

such that a = b * k. And so we can now look at the number.

If we are interested in c * a,

and it is equal to b * c * k. And again,

it satisfies the definition of divisibility by b.

So, b / c * a, as well.