1 00:00:00,000 --> 00:00:03,500 The most important definition in this course. 
It’s weird that I'm ending a video with this, 2 00:00:03,500 --> 00:00:06,100 and it's weird that we ended 
a week with this definition, 3 00:00:06,100 --> 00:00:10,200 but seeing this definition several 
times and having a… you know… 4 00:00:10,200 --> 00:00:12,766 even the weekend, or a couple of days, or if it's the 5 00:00:12,766 --> 00:00:15,500 reading week, having the reading 
week to practice this definition 6 00:00:15,500 --> 00:00:17,366 is invaluable to you. 7 00:00:17,366 --> 00:00:20,800 This is something - I can't stress this 
enough. You have to know this definition. 8 00:00:20,800 --> 00:00:23,733 The rest of this course is 
going to use this definition 9 00:00:23,733 --> 00:00:27,133 like it's child's play, like it would 
use an equal sign, okay? 10 00:00:27,133 --> 00:00:30,233 So make sure you understand this 
definition, I can't stress this enough, okay? 11 00:00:30,233 --> 00:00:33,366 Make sure you know this. 
"Let m be a natural number. 12 00:00:33,366 --> 00:00:36,833 We say the two integers a and 
b are congruent modulo m 13 00:00:36,833 --> 00:00:39,800 if and only if m divides 
their difference.” 14 00:00:39,800 --> 00:00:42,400 Okay? It's a very 
simple definition, 15 00:00:42,400 --> 00:00:45,466 but it has far-reaching 
consequences, okay? 16 00:00:45,466 --> 00:00:51,000 We denote this by writing a is 
congruent to b mod m, or modulo m. 17 00:00:51,000 --> 00:00:56,000 If m doesn't divide a minus b, then we write 
a is not congruent to b mod m, okay? 18 00:00:56,000 --> 00:00:58,233 19 00:00:58,233 --> 00:01:01,266 This is the definition. Okay? 20 00:01:01,266 --> 00:01:03,566 Learn it - commit it to memory. 21 00:01:03,566 --> 00:01:06,933 If a is congruent to b mod m, then 
m divides the difference of a and b. 22 00:01:06,933 --> 00:01:10,400 Just that’s…you have to just 
look at that and know that. 23 00:01:10,400 --> 00:01:13,133 It needs to be obvious, 
it needs to be quick. 24 00:01:13,133 --> 00:01:15,500 So when you're doing 
things like the following: 25 00:01:15,500 --> 00:01:17,466 like 3 is congruent to 7 mod 4, 26 00:01:17,466 --> 00:01:20,700 you just need to look at this and be like, “oh 
yeah 3 is definitely congruent to 7 mod 4. 27 00:01:20,700 --> 00:01:23,300 4 divides the difference of 3 and 7.” 28 00:01:23,300 --> 00:01:25,300 10 is congruent to minus 8 mod 9 29 00:01:25,300 --> 00:01:28,933 of course, right? 10 minus 
negative 8 that's 18. 9 divides 18. 30 00:01:28,933 --> 00:01:29,666 31 00:01:29,666 --> 00:01:32,133 4 is obviously congruent 
to 4 mod anything. 32 00:01:32,133 --> 00:01:34,000 mod 6, mod 10, mod 15. 33 00:01:34,000 --> 00:01:35,233 34 00:01:35,233 --> 00:01:37,800 It needs to be quick, okay? 
You need to look at these - 35 00:01:37,800 --> 00:01:40,900 even for larger numbers after a little 
bit of practice, you'll be able to say, 36 00:01:40,900 --> 00:01:43,800 “Oh I know that 3 is 
congruent to 43 mod 4,” 37 00:01:43,800 --> 00:01:46,600 and that won't seem 
like anything to you. 38 00:01:46,600 --> 00:01:48,400 I'll know that, 39 00:01:48,400 --> 00:01:52,300 you know, 10 is congruent 
to 9001 mod 9. 40 00:01:52,300 --> 00:01:54,800 These are things that you'll 
know, and after a little bit of 41 00:01:54,800 --> 00:01:56,033 42 00:01:56,033 --> 00:02:00,000 practice, you'll get used to why 
these things are true, okay? 43 00:02:00,000 --> 00:02:01,100 44 00:02:01,100 --> 00:02:03,266 But you do need to 
know this definition. 45 00:02:03,266 --> 00:02:06,766 Spend some time with it, play around 
with it, try to come up with some examples. 46 00:02:06,766 --> 00:02:09,533 Make sure you memorize 
it, make sure that like 47 00:02:09,533 --> 00:02:12,866 once this video is over, which 
will be in about a minute or so, 48 00:02:12,866 --> 00:02:15,600 that you can sit down and 
write down the definition. It's 49 00:02:15,600 --> 00:02:18,766 good to practice your recall. 
Again, this is important. 50 00:02:18,766 --> 00:02:20,766 It's just important. 
Learn this definition. 51 00:02:20,766 --> 00:02:23,800 For the next 3, 4 weeks 52 00:02:23,800 --> 00:02:27,300 possibly to the end of the course, this definition will appear in many ways, 53 00:02:27,300 --> 00:02:30,400 and we’ll expect students 
to be comfortable with this. 54 00:02:30,400 --> 00:02:32,833 It's a very important 
definition in Number Theory. 55 00:02:32,833 --> 00:02:33,566 56 00:02:33,566 --> 00:02:36,900 So that's all I have to say about week 6. Again just a quick summary: 57 00:02:36,900 --> 00:02:40,200 we talked about gcd concepts, we talked 
about Linear Diophantine Equations, 58 00:02:40,200 --> 00:02:42,566 and we introduced congruence. 59 00:02:42,566 --> 00:02:45,866 Again, in this video, I didn't talk 
about exactly why this is important 60 00:02:45,866 --> 00:02:48,100 or what we need this for. 
We'll see that in week 7, 61 00:02:48,100 --> 00:02:50,733 I'm going to save that until later. 
The real most important part of this 62 00:02:50,733 --> 00:02:55,200 week is learning this definition, and making sure 
that you understand at least some basic examples. 63 00:02:55,200 --> 00:02:58,966 That's it. Thank you very much for 
your attention and best of luck.