The most important definition in this course. It’s weird that I'm ending a video with this,
and it's weird that we ended a week with this definition,
but seeing this definition several times and having a… you know…
even the weekend, or a couple of days, or if it's the
reading week, having the reading week to practice this definition
is invaluable to you.
This is something - I can't stress this enough. You have to know this definition.
The rest of this course is going to use this definition
like it's child's play, like it would use an equal sign, okay?
So make sure you understand this definition, I can't stress this enough, okay?
Make sure you know this. "Let m be a natural number.
We say the two integers a and b are congruent modulo m
if and only if m divides their difference.”
Okay? It's a very simple definition,
but it has far-reaching consequences, okay?
We denote this by writing a is congruent to b mod m, or modulo m.
If m doesn't divide a minus b, then we write a is not congruent to b mod m, okay?
This is the definition. Okay?
Learn it - commit it to memory.
If a is congruent to b mod m, then m divides the difference of a and b.
Just that’s…you have to just look at that and know that.
It needs to be obvious, it needs to be quick.
So when you're doing things like the following:
like 3 is congruent to 7 mod 4,
you just need to look at this and be like, “oh yeah 3 is definitely congruent to 7 mod 4.
4 divides the difference of 3 and 7.”
10 is congruent to minus 8 mod 9
of course, right? 10 minus negative 8 that's 18. 9 divides 18.
4 is obviously congruent to 4 mod anything.
mod 6, mod 10, mod 15.
It needs to be quick, okay? You need to look at these -
even for larger numbers after a little bit of practice, you'll be able to say,
“Oh I know that 3 is congruent to 43 mod 4,”
and that won't seem like anything to you.
I'll know that,
you know, 10 is congruent to 9001 mod 9.
These are things that you'll know, and after a little bit of
practice, you'll get used to why these things are true, okay?
But you do need to know this definition.
Spend some time with it, play around with it, try to come up with some examples.
Make sure you memorize it, make sure that like
once this video is over, which will be in about a minute or so,
that you can sit down and write down the definition. It's
good to practice your recall. Again, this is important.
It's just important. Learn this definition.
For the next 3, 4 weeks
possibly to the end of the course, this definition will appear in many ways,
and we’ll expect students to be comfortable with this.
It's a very important definition in Number Theory.
So that's all I have to say about week 6. Again just a quick summary:
we talked about gcd concepts, we talked about Linear Diophantine Equations,
and we introduced congruence.
Again, in this video, I didn't talk about exactly why this is important
or what we need this for. We'll see that in week 7,
I'm going to save that until later. The real most important part of this
week is learning this definition, and making sure that you understand at least some basic examples.
That's it. Thank you very much for your attention and best of luck.