Well as I stressed in the lecture, you've got to be careful in distinguishing
between these two notions. You have a notation for a relationship,
something that's true or false. A is divisible by a or b divides a.
And you've got another notation for a number, the result of dividing a by b.
This makes sense when you're talking about the integers.
This actually is not defined for the integers.
I mean, for some integers, you get an answer.
But this is a s- -- I mean, division, actual division as an operation, is an
operation not on the integers. It's an operation on the rational numbers
or the real numbers. (End of transcription.) in, in the
integers all you have is addition, subtraction, and multiplication.
You don't have division. What you can do is say whether you have
divisibility. And divisibility is define, defined in
terms of multiplication, okay. So it's all about the distinction between
the 2, you've got a property, assuming it's true or false And you've got a
notation for a number. This is an actual number.
Okay. So the actual answer I'm going to end up
with is the thing here at the bottom, to the question.
This concisely and accurately as you can the relationship.
This is the relationship. B divide into a or a is divisible by b if
and only if a divided by b happens to be an integer.
And so the, summarizing what I just said above there, that's a notation that
denotes a rational number, rational number has a different set of numbers.
This denotes the relation, B divides A, i.e, there is an integer Q, such that A
equals QB. And whenever you're dealing with
divisibility, with this notion and you're working with the integers, you have to
reduce that abbreviation. To this.
There is an integer switch at that. That's what that means.Okay?
That means that. That little thing means that.
Okay? And then to get down to here in the case
where you do have divisibility, then of course, the q that's here is the quotient.
I mean we use q to stand for the word quotient anyway, but notice that this says
nothing about division. Division doesn't arise here.
It's all about. The results of multiplying two numbers.
So this makes perfect sense when you're talking about the integers.
Okay? So, we're not doing, sort of, arithmetic
in the sense of calculating here. You know, obviously, everything that's
involved here about dividing one whole number by another.
You know, even[INAUDIBLE] in elementary school, it is just division of whole
numbers. The focus here however, is on what you're
doing within certain systems of numbers. We have 2 systems of numbers here.
We have the integers and we've got the rationals.
They're just 2 separate systems of numbers.
In the case of the integers, you can add, subtract and multiply.
In the case of the rationals, you can add, subtract, multiply and divide.
But they're different systems of numbers. And so the focus here is on what you can
do with the integers. That's what number theory's about.
In then in the very last lecture, lecture ten, we'll be actually looking at the
rationals and the reals. But that's a different system[INAUDIBLE].
Numbers. You can do different things with it.
So we're taking a more sophisticated look at elementary arithmetic.
But it still is, after all, elementary arithmetic.
Okay, let's look at numbers two and three. But the issue with all problems like this
is you have to express the divisibility property in terms of multiplication.
Remember, divisibility is a property of pairs integers.
You can't divide integers, all you can do with integers is add, subtract or multiply
them and divisibility arises when you Take this definition, a divides b if and only
if there is a q, an integer, such that b equals q times a.
So the general strategy for dealing with divisibility or here, seems to be the only
strategy, is you replace issues like this, you replace a statement like that With a
statement about multiplication. Because the point is that there is no
operation here, there's nothing, that, that's not an operation to do, there's not
an arithmetical operation on the integers. And remember this is all about the
integers. So you have to express it in the en, in
the language of the integers. And the language of the integers allows
you to talk about addition. Subtraction and multiplication, but not
division. Okay, so how do you show that I mean, how
do you answer this one? What's the proof?
Well, this one is, is sort of immediate because the, the very definition of
divisibility explicitly excludes A not being in 0.
A not equal to 0. It excludes A being 0.
Okay? So it's false, and that's the reason.
9 divides 0? Well that's definitely true.
And to that show it's true, you simply express.
The definite, you can express it in terms of the definition.
So you would have to show that there is a q, look at the definition, you have to
show there is a q, so it's at 0, equals q times 9.
Well of course there is, 0 itself is one of those things.
Okay so that's false. It's true, this one's is false, for the
same reason a was false. You're not allowed to, to have a equal to
0 in, in the notion, that includes that requirement.
This one is definitely true, and the proof is just write it in terms of this.
This is basically what you end up having to write.
In each case, if you look at these, that's what I'm going to end up having to write.
I'm going to end up reformulating it in terms of the definition.
That's really all it involves, just reformulate the statements.
In terms of the definition. Sure that the definition is true.
Okay. Well in this case it's true because q
equals 1 makes it true. In this case, we know that there's no such
q. I mean, you could argue it just by since
any possible q would have to be less than what, less than 7, say.
You could actually, explicitly. If you wanted to prove to that even more
detail. You would just let q be all of the
possibilities that have a chance of being that.
Q equals 1, 2, 3, 4, 5, 6, 7. And 7 times is already 49.
So, you're, you're, you're out of it. So, you actually only need to go to 6, of
course. So You could explicitly list all of the
possible Qs if you wanted to. But that would be so trivial.
I think you could just leave it like that. At this level, if this was, if we were
talking to kids in the elementary school, we would ask them to maybe list all of the
possible Qs and make sure that none of them give you the answer 44.
But at this level you, you can just take that out That's been obvious, alright?
This one's certainly true. You exhibit the Q, now it means Q equals
minus six, Q equals negative six. Ditto here, you exhibit the Q and again,
it's, it's negative seven. Here you exhibit the Q.
And the q is 8. Here, you need to show that for all n 1
divides n. Well, that's certainly true.
And the reason is that for any n in Z, n equals n times 1.
Right? [laugh].
That's trivial. Right?
1 divides everything. Fallen in Z, fallen in N, N divides 0,
that's true, because, again, for any N in Z, 0 equals 0 times N.
And this one, this is one we've gotta be careful with, because if we're quantifying
over all of the integers, that includes Zero itself.
And you're not allowed to have zero dividing anything.
Okay, that's excluded from the definition. So this is the one you have to be careful
with, because it includes. Zero.
Its not a case where it goes wrong, but you only need one counter example to make
unifunds, to make universally quantified statement false, and that one counter
example is all it takes to get rid of that one.
Okay, so that one is false. Okay?
Now we've done them all. Notice it was just a same pattern, express
divisibility in terms of the definition of divisibility, and then each case it just
drops right out/g. Because this is afterall just elementary
whole number arithmetic, you know, it's not that there's anything deep going on
here. It's just that we're looking at it in a
somewhat more sophisticated fashion than you did when you were the elementary
school. Everything you need to know to solve this,
you learnt in elementary school. It's just that we've now a little bit more
of a sophisticated gloss on it. Okay?
Well, as with the previous example, all you have to do is reduce each of these to
the definition of divisibility. Remember divisibility is a property of
pairs of integers, this isn't division. It's obviously related to division but you
don't have division In the integers. What you can do with the integers is you
can add them, multiply them, and, and subtract them.
I mean, subtraction just being you know, the inverse of, of addition.
But you can't divide them. Okay?
But you've come to have a property of divisibility, but to discuss divisibility
within the integers, you have to reduce it To, to a discretion of essentially
multiplication. Okay so, how would you show that a divide
0, that, that's you got divisible [INAUDIBLE]by a, well.
You observe that actually, because of the properties of 0, 0 is equal to 0 times a,
so in particular, 0 satistifes the requirement for divisibility.
Okay? There is a q in z, so it says 0 equals qa,
namely, q equals 0. So, by definition, a divides 0.
Okay, similarly, in the case of a dividing a, Because of the properties of, of 1, a
equals 1 times a. So again, the definition of divisibility
is satisfied. And it's satisfied in this case, by
letting q be equal to 1. So by definition, a divides a.
Okay. So that was that one.
And the rest are essentially the same idea.
A divides 1 only if a equals plus or minus 1.
Okay, well we've got two implications to prove.
First of all, let's assume that a equals plus or minus 1.
Then again, all you have to do is show that there is something so it's that one
equals Q times A. Well, if it equals plus or minus one, it
certainly is right? Conversely That should be if, little typo
there. Conversely, if a divides one, then for
some q, one equals q a, by definition of divisibility, but if one equals q times a,
then the absolute value of one is the absolute value of q a, which is the
absolute value of q times[UNKNOWN] the value of a And if 1 equals that, then the
only possibility because these are positive integers now, is that absolute
value of q is absolute value of a is 1, that's the only way you can get 1.
And so if the if the absolute value of a equals 1, then a has to be plus or minus
1. [inaudible] 1.
Let me just do one more and then let you to do all of the rest.
If a divides b and c divides d then ac demands bd.
Okay? Well we know that there are q and r so
it's a b is qa definition of divisibility. D equals r c definition of divisibility
hence multiplying the two together you've got b d is q r times r c which is when you
rearrange them q r times a c. So by definition a c divides into b d.
And the others are essentially the same idea.
In each case you just reduce it. To the question of, of multiplication
through the definition of divisibility. So you never actually do any dividing, you
express division in terms of multiplication.
And you can do that because division is the inverse of multiplication.
Okay? So it, the whole thing is going to work
out. So these proofs are always typically just
1 or 2 lines. They're really just a matter of
translating what it is you're having to prove into divisibility.
So the 1st line of any of these arguments really is just a matter of re-expressing
what it is you're having to prove. In terms of divisible, in terms of
multiplication. By the definition of divisibility.
Okay. Well, that's it.
