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Summary. We did a lot 
of theorems this week.

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An absolute ton. I didn't even prove them 
all, I didn't have them all in this slideshow.

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I actually didn't include 
Finding a Prime Factor,

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right, that a prime factor must be less than 
the square root of the number if it exists.

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So if a number’s not prime, 
if a number’s composite,

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then it has a prime factor that's less 
than the square root of the number.

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That's FPF,

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again, I didn't think it’s as 
major as the other theorems,

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but there's a ton, a ton, a ton of
theorems in this course, okay?

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Even just this week alone had like 8 or 9 
or whatever was, right, there's just a ton.

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There are theorem cheat pages, they are
available on the Math 135 Resources Page.

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Practice so… 

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do I like using these? I mean,

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yes or no. You should practice with 
them at the beginning, it's hard,

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but you should also practice 
without the cheat sheet.

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Why should you 
do that? I mean

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in many offerings of this 
course, they do give you -

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or they have given you - 
a cheat sheet on the final,

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so if you do or don't get a cheat sheet 
on the final, that shouldn't matter.

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You should still practice with 
and without the cheat sheet.

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By practicing without the cheat 
sheet, what are you doing?

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You're forcing yourself to actually 
remember what you did in the week,

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and by trying to remember 
what you actually

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did in the week is actually going to help 
you long-term remember the content,

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and remembering the content is
really what you should be trying to do.

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You shouldn't be trying to 
cram for a test. That's

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definitely not the
philosophy of Math 135.

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This is a course that you're 
going to use the core ideas for

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in almost every single 
math course you take.

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You really want to make sure 
that you know these ideas,

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and you have them
down solidly, okay?

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This is not something you 
want to goof around with, and

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it’s not something you want 
to just cram and forget.

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Math, in general, is not something
you want to cram and forget,

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but this course especially will hurt 
you long-term if you do that.

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So practice. Practice is important,

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practice recalling theorems
with and without the cheat sheet,

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practice in different settings, practice with 
multiple questions, practice the first, you know,

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mix in a problem from week 
3 and try some gcd problems,

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practice with mixing is

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absolutely vital towards 
success in this course, okay?

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Don't just pocket your 
theorems. Try to mix them up.

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That's what this 
is trying to reiterate.

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The toolbox analogy. Again, 
if you get stuck, just

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calm yourself down, don't 
panic, panicking is always bad,

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and just ask yourself, “Okay.
Do I understand the question?

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That's true, then what tools do I 
have to answer the question?

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Oh, well I have GCDWR, does that
apply here?” Well GCDWR applies

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usually really well 
when the gcd…

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when the parameters in the 
gcd depend on each other.

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So if that's the case, then using GCDWR 
a couple of times will help you out.

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You know, if that didn't 
work, well maybe

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there's a gcd condition in the
hypothesis and I should use

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there's a gcd condition in the
hypothesis and I should use

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Bézout’s Lemma, or Extended 
Euclidean Algorithm.

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Maybe the gcd is in the 
conclusion, I can use GCDCT,

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I mean you have a toolbox. 
Go through it step by step

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and ask yourself, “Which of these tools 
makes sense to use? Am I going to get closer

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to my goal by 
using this tool?”

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So hopefully that gives you just a

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brief outline of the gcd content.

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Again this is a really packed week,

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so this video is probably a 
lot longer than I want it to be.

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I really want it to be 10 minutes 
and I'm sure I'm over 30 right now

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but hopefully you do it bit 
by bit, piece by piece,

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and get a little bit out of it…
out of what you're doing.

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So thank you very much for your 
time. Thank you for listening,

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hopefully this helped
you out and best of luck.

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