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Last topics of the 
week were divisibility.

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I know that some people didn't get 
to divisibility this week, that's okay.

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Here's an interlude 
to next week.

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We say that m divides n 
and write m vertical bar n,

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we read this as m divides n,

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if and only if there 
exists an integer k

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such that 
m k equals n.

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So I told my class 
this “and only if” part,

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we'll get to that 
next week, okay?

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Usually, mathematicians 
don't write this,

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they're generally sloppy

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with their definitions 
in this sense, but

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it should be “if and only if”. I'll try to do that
throughout the course for myself,

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but note that other 
books might not do this.

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Otherwise, we write m 
does not divide n so

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m vertical bar 
slash, this is

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“slash not slash mid” inside 
LaTeX if you're interested.

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That is when there's no integer k 
satisfying m k equals n.

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That's just the
definition of divisibility.

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 From here…divisibility is a very 
core concept in this course.

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We kind of understand divisibility 
from an intuitive standpoint, right,

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like 3 divides 9,
negative 6 divides 36,

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the only ones that are weird 
are things like 0 divides 0…

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remember 0 times 3 is 0, right? So it 
satisfies the definition of divides, right?

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Same with things 
like 3 divides 0

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because 3 times 0 
equals 0, right?

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So just keep that in mind
that 0’s can be in here,

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just remember
which way they go.

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So if m is 0 and this is true, 
then n also has to be 0.

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I'll let you think about 
that for a minute.

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Finally bounds by divisibility, this 
was the last major thing we did.

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This is the first look at 
an acronym in the textbook.

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I also like to use DeMorgan's 
Law, DML, as an acronym but

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BBD is the first one in the 
book, bounds by divisibility.

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How do we read this? So we have a 
divides b and b does not equal 0 implies that

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the absolute value of a is less or 
equal to the absolute value b.

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So one thing I did 
here intentionally...

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unlike before...one thing we 
do here intentionally is if

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we don't specify the 
domains of these symbols,

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then take them to be as 
big as they can possibly be.

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So because we defined the
vertical bar for integers,

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we're gonna take a and b as integers. 
That's really what I want here anyways,

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but I want to make this note because 
it is possible that I will forget

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during class, or I will 
forget during…something…

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during an exam, hopefully the 
questions are written crystal clear,

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but you know 
mistakes do happen.

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But the exam should be clear
otherwise, take that to mean that.

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Same with other textbooks as 
well. This is usually a golden rule.

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What's the proof of this? Well okay, 
if I take two integers a and b

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satisfying the two
requirements,

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then the definition of a divides b, 
remember that's all we really have, right?

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So when all you have are hammers and nails, 
then you should probably nail hammers…

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or hammer nails…
I got that right…

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You should probably 
hammer nails, right?

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All you have is a 
hammer, use the hammer.

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So all we have is the definition,
so let's use the definition.

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So we know there's an
integer k such that a k equals b.

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Since B does not equal 0, what 
do we know? We know that

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K is non-zero and so 
we have the following 

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sequence of statements.

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So the absolute value of a 
is less than or equal to

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the absolute value of a
times the absolute value of k.

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that’s true because k is 
non-zero and k is an integer, so

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the absolute value 
of k is at least 1,

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so therefore 
this holds.

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Remember absolute values, 
you can take them in,

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so now I have the 
absolute value of a k.

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That's just equal 
to b, as required.

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Last thing I 
want to mention,

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a lot of you probably already 
know this but, I do have

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symbol and theorem cheat sheets on
the Math 135 Resources Page

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so, for example, I didn't 
mention here but

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Z being the integers,

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Z being the first letter of 
the word integers in German,

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there exists is 
this backwards E,

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we’ll see this a lot 
this upcoming week,

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but I want to introduce 
it now just so that

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it's a little bit easier to 
cope with these symbols.

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You're gonna see a lot of symbols 
over the next couple of days,

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the beginning of this course does
have a lot of symbols, but

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once you get them down it's a lot 
easier to write things mathematically.

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There are theorem 
cheat sheets.

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So the first couple of weeks 
the theorems aren't too too bad,

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but I did include them.

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After week 
five or six,

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the theorems start to
pick up very heavily,

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and there's where the cheat sheets 
will probably be more useful to you.

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But, you know, 
for the moment,

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the symbol cheat sheet 
might be more useful.

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Again though, there are theorem
cheat sheets to help you out.

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Okay, so again thank you 
very much for listening.

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Hopefully this gave you a 
little bit of an insight as

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to what we did, 
where we're going,

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why these things 
are important.

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Again if you have any 
feedback, email me at

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cbruni@uwaterloo.ca

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post on Piazza,

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and that's it. Thank
you very much.