 I think it's good isn't it okay okay wait a minute okay is this thing does this normally it's good right that guy in the back probably would have said something yeah I think it's fine it's good okay so here's what we're going to do I will hand your exams back I'll do actually I'm going to do that in the beginning of class today and then and we'll talk a little bit about this the scores are pretty good you probably found that this is probably the easiest out of the three exams based on your scores I'm guessing or maybe you just got worried you're going to fail and you started studying harder I don't know but overall yeah it's pretty good I'm going to post solutions like I normally do probably today but if not today then they'll be up tomorrow so we'll talk about that briefly and then we will talk a little bit about the the final I'm going to go and do a couple of problems talk about a couple of your homework problems I actually because last week such crazy week I think we're just going to kind of finish off with what we've done last week so I know probably some of you feel like you're not getting your your money is worth here but you've done enough work enough homework I think you probably found this to be somewhat challenging class and maybe you've honed your logic and proof writing skills a little bit and that was the main goal anyways really it's not to really come away with a really advanced knowledge of number theory because that stuff gets crazy after a while but hopefully with those of you that are going to go on and take abstract out where you have a lot of experience now writing proofs and maybe you've gotten a little bit better at that and you know but I've that's what I'm going to do we're not going to go into anything else today so final exam just give you the basics here I'm not going to talk about exactly what's what's going to be on it yet but just to recap with the guy that was here last week probably I wanted to tell you a few things hopefully he did that sideburns oh yeah yeah well he has I don't have sideburns so I guess by default I guess that sort of is true but yeah I was giving brief reason his big sideburns but so he told you that the final he gave you a breakdown of where the problems are going to be coming from did he give you this okay okay can somebody remind us all what that was okay yeah that's right okay so half the exam stuff that you will have already seen before I think that's pretty fair given that this is a comprehensive exam but of course we haven't done that much really this semester so solutions are all going to be you know posted you all know where to find these by now hopefully and then you know the other the other half is just going to be stuffed that you you know again should should be comfortable with I think you found on the last exam that the problems were not anything weird I mean these are all stuff things that you've seen before more or less you know so it's going to be the same thing on the final exam it's going to be weighted out someone's going to ask okay is it is going to be more heavily weighted towards the material you know toward the end and now the answer is no everything it's really going to be weighted about the same so you should just be looking over everything look over your debt you know the definitions look over the things I told you look over for the for the three tests and you know aside from that make sure that you know what you did wrong on the exams make sure you know what you did wrong on the graded homework and then you know aside from that I would just say just practice practice some problems and you know just make sure that you know what you're doing and next week of course we don't actually have class next Tuesday right so what I will do is on Thursday I'll let you know what I'm going to be available next week so I will still be around even though it's not and maybe it'll be during class normal class time I don't know yet so I'll announce that but I will definitely be around next week and we can talk about some stuff you just want to come to my office we can talk about whatever so yeah that's that's basically I guess I should say the exam is what from 140 to 410 I think it is somebody make sure I'm right about that okay yeah 140 to 410 okay that's next Thursday next Thursday and we really sort of have to follow this the schedule so but of course that gives you plenty of time to study one thing I'll mention and you guys are pretty good about this but I always say it anyways it's going to be it's going to put you in kind of a tough position if you all of a sudden get sick on on Thursday can't take the exam well yeah or that yeah if you're going to do that I would definitely prefer that you be out of the classroom that maybe that's a little distracting so yeah that obviously if you really are sick or something happens yet of course will accommodate you but I'm going to be probably going out of state for a couple of months and I'm heading out here as soon as I can so I we can we can still work it out but that's gonna you know don't be here if you can't that's all I'm saying okay it's definitely in your best interest because otherwise you're gonna have to wait until until August and then you're gonna have all this time you know where you you know open out the amusement parks and you've been getting baked in the sun and baked from other stuff and then by the time you come back here just grains of mess so yeah so why don't I just go ahead and get your exams back I think that lamb boy told you that I added five points to your scores right okay so you'll see you'll see the you know how many you got so there are 75 points total I just automatically added the five points you say oh that's not right but it's with the five points added your percentages with the five points at it even though I didn't I get too lazy to write it down so okay a lot of stuff actually yeah I'm I'm got no I'm getting recorded now I can't I cannot say anything maybe the before class on Thursday I can go into some details okay Eric Josh always makes you guys up okay yes yes okay do you want to give this to your friend David and your Sarah right where's Chad back here and Michaela passing that down to you okay Andrew and I think that's everyone okay good okay so I'll say a few things about the exam I'm not going to write very much down but because it'll be posted online let's see okay yeah I mean number one you should have gotten because it came right off the last exam so definitely better than it was but I could tell also that some of you did not look over this one and number two to be is actually false a lot of you confuse that with the converse the face congruent to b mod n then a squared congruent to b squared mod but the other direction is not true in fact you can see if you think about it I'm not actually even going to give you a specific example think think about it this way what do you what do you know if a squared congruent to b squared mod n that means that n divides a squared minus b squared which factors as a plus b times a minus b so if you know that n divides a plus b times a minus b the really the question just becomes this if n divides a plus b times a minus b does n divide a minus b then odds are that's probably not true right if it divides a product does it necessarily have to divide the first factor no that's not going to be true in general so that was false okay 3a was just the type of problem that we've talked about before so I mean the idea here I think was just the notice that two to the third is eight and that's congruent to one mod 7 and once you've got that it's pretty easy after that right okay number 3b overall was actually I thought this was going to be a disaster but it was it was not as bad as I thought it was going to be and of course we talked about things similar to this you had similar problems at least one or two in your homework right so yeah I don't think I'm going to say much more about that 4a most of you got that 4b most of you got that and yeah let's see just trying to see if there's anything I really need to point out here number 5 this problem was actually in the book and this is one of the problems I actually did in lecture when we talked about the Chinese or maybe doing so if you missed it chances are you you just made a silly computational error or you just didn't study enough how to do the algorithm because it's just an algorithmic problem I mean there's not much to it right okay number six finding solutions mod 21 this also it's just a standard problem you check to see if the GCD of 6 and 21 divides 15 GCD is 3 that certainly divides 15 so there are three solutions right and then from there it's not really hard to solve I think the easiest way to solve it is just to go through and divide everything by three at first and then you get one solution then you have to add n over d so you have to add 7 twice get all three of them right okay and the last one this some people had some some issues with this one although it was really not much different than what I've talked about in class now I could reduce the 9 of course to 1 but I wrote this problem specifically because I wanted you to recognize that there are a couple ways you can go with this right you can bring the x term over or you can bring the y term over notice that the coefficient in front of y is and 8 the modulus are not relatively prime so you're sort of if you really want to do it the way the rules dictate you have to you really need to bring the 2 y over to the other side because 3 and 8 are relatively prime right and once you've got that you know that there is a unique solution no matter what for so again for any value of y there's a unique solution modulo 8 for x and then you just go through it the way I did it in class really okay so this is definitely something I did and we talked about this right so you do it if you do it the way I would have done it you would end up with y equals t I think x was something like 3 6t plus 3 or something this person didn't get it right so but yeah and that was that was it really so I think I think you guys got a fairly reasonable exam here and so I will say that especially with the extra credit the 5 points extra credit with that awesome extra credit problem I put on the second exam probably there's not going to be a whole lot of a curve here especially with me giving you half the exam being stuff you've already seen before okay so don't expect there to be some big curve where you know 65s in a that's not going to happen so no not gonna happen so no I if you if you have if you have your baby between now and the final I will give you I will give you some extra credit okay because that would be a ordeal to have to come to show up the final so yeah you have to show up no matter what by the way so I changed my mind okay so let's see okay so here's also what I'm going to do the homework that okay so you have a homework that you'd already turned in that you need to get back okay and then there's this final assignment section 5.2 so what I'm going to do is I'm I'm going to give you I'm going to it's actually I'm going to extend the the due date of this last assignment to the final okay what I'm going to do is between today and Thursday is we'll just talk about a lot of the problems okay and so yeah I mean I'm I'm going to give you a good amount of help with this you know but I so that what that means is that you know yeah the solutions are not going to be posted yeah so that but that so that just means that's going to eat into the 50% of the other stuff basically yeah okay so I want you guys to ideally we'll talk about a few things but would since we only have one more class period to think about you know really really work on these I'm not going to I'm going to go back so here's what we're going to do today okay I'm going to talk about a few of these homework problems I'm not going to talk about a lot of them yet because I really want you to spend some time processing and trying to think about it we'll talk about more on Thursday and then I'm going to go back over and sort of remind you of some things that you should know especially things that I noticed that some of you had a lot of trouble with and then Thursday will do you know maybe I'll do a couple induction problems you know maybe some proofs from earlier maybe that you've forgotten about a little bit because things have gotten a lot more computational in the last couple sections then we'll you know talk a little bit more about the homework so that's that's sort of the plan okay so you know where the you know where to study right because you know half of the exam is coming from stuff that we've already seen so probably not probably not because you you don't need one so yeah I think and I also you know want to see that you guys can you know do things like multiply 52 times 4 without a calculator no your your concession is your concession is the is the that 50% of the exam is stuff that you've already done that that's a pretty big concession yes yeah you want to if you want to email me yeah if anyone you know yeah if you want to ask questions about your grade feel free to email me and I'll let you know what I can tell you is is this if you okay so each exam is 15% of your grade okay and the let's see what was the homework is it 35% okay yeah the finals yeah okay right the finals 20% okay so here's what you can do if you want to get an idea your homework let's see yeah I'm definitely going to drop the lowest homework I haven't decided if I'm dropping any more than that but I will definitely drop one for sure okay so what you can do is just just do this I mean if you want to get an idea where you stand take your first exam score whatever whatever that percentage is take that percentage of 15 do the same for the the second the third exams and if you average your homework just take that percentage of 35 then then you get a you know then we add all those numbers up right you'll divide by 80 and then that's where you're at basically okay so we're in a number theory class I would think you should probably be able to handle that so yeah I mean if you you know if you're missing some homework or you know of course I have them up here but you guys really should be able to figure this out yourself really okay but of course if you have any quite I mean I'll help you but you know you should be able to do most of it yourself okay so let me just start by reminding you of a few things I think would be important for you to maybe practice okay that I've seen a lot of mistakes on okay induction should definitely be able to do a proof by induction and that's kind of what where we started in this course was induction so I would strongly encourage you to look over that and you know there was also there's still some issues with even writing the statements of the first principle and the second principle of induction so what just happened I guess I haven't done anything for too long right okay all right let's see what we have to do here oh so that was not the right thing oh yes it worked okay good I did not expect that to work but it okay wow okay 1.2 binomial theorem like I was saying you should know the statement you should also be able to do some proofs with the binomial theorem and the plan is you know I'll remind you of a few of these we'll do a couple of these maybe today or on Thursday okay so that's was it for chapter one chapter two we started with the division algorithm so you should definitely know what that is I could ask you state the division algorithm I think I did that actually on one of the exams and so again you know this is a kind of a formal mathematical statement right so it just says that if you take any integer well it's it's in the book but if you take any integer a and positive integer n there exists unique integers q and r such that a equals nq plus r and r is between 0 and n one's inclusive one's not right can be 0 but it can't get to n so you shouldn't be writing things like most of you didn't do this a few of you did you shouldn't be saying something like you divide by n and then there's a remainder and that's not what it says okay okay I don't think you did that let's see okay all right so and then we started in with divisibility properties and these are things that you know now we've been using these a lot and we're talking you know we've been talking about congruences which is just another way of expressing a divisibility assertion yeah there's not a whole lot for me to say here you should you should just be comfortable with it you should at least be comfortable with with using things like you could look excuse me Euclid's lemma right you don't have to some of these things you're not they don't have names so I'm not going to ask you specifically to define them or to state the theorems but you should be able to work with them so for example if you know that a divides bc and a and b are relatively prime than a divide c so you might have a problem or proof where that fact is going to be used and so you know you should definitely know what it is even if you don't call it by name but you should be able to work with it for example okay we skip just to remind you the Euclidean algorithm we we actually talked a bit about that section but we didn't do the Euclidean algorithm itself this is basically we started talking about more properties of the gcd and then the least common multiple okay then two five we we did not do and let's see three point one okay primes so this is the fundamental theorem of arithmetic this kind of stuff okay so you should definitely know for example what that says I will not ask you to prove it but you should know what it says you should be able to do things like prove that the square root of two is irrational I think the problem you had to do in this section was to show that let me see maybe not at some point anyways I'm pretty sure I asked you to do the problem that asked you to prove the square root of p is irrational p is a prime so you should be able to do that as well and then obviously goes without saying the homework you should be comfortable with because some of these are going to be on the exam okay and then yeah that was in three two actually and yeah I don't have a lot to say about three point two Euclid's proof that they're an infinite number of primes appeared in that section and you should just be comfortable with the stuff that we did 33 Goldbach conjecture we did not talk about dear slaves there oh I didn't tell you about that yet man I have not been a good professor this semester okay so then we moved on to chapter four with congruences we started with a basic stuff in four two and then we skip four three and four point four linear congruences and the Chinese remainder course this is recent stuff now so this is the kind of stuff we did the last exam should be comfortable with congruences and how to solve those Chinese remainder theorem of course we've just been talking about that and then the last thing is five two with Fermat's theorem and then a couple of corollars and that is yeah that's about it so what we'll do is you know today I may go back and do an induction problem and then I'm going to talk about a couple problems in five two today like I said I'm not going to I'm not going to talk about it a lot of them so I really want you to spend a little time if possible between now and Thursday thinking about some of these yourselves so I'll talk about a couple maybe two or three and then we will do an induction problem binomial theorem problem you know something else you guys feel rusty on you want to see you know I can do pretty much whatever you guys want me to do so let's just do that first though to make sure we get that out of the way so let's see the yeah the web page at least with the assignment will be updated today for sure I'm going to try to get the solutions posted as well but like I said if not today then they'll be out by class by Thursday's class okay so I've forgotten how to do this okay five two there we go I do want to say something I may not do this whole thing but this whole problem but number one ask you to show that 11 to the 104th power is congruent to minus one mod 17 okay so here's the idea well actually that's not what it said sorry I okay it said that I mean yeah basically it said I to prove that 17 divides 11 to the 104 plus one right that was the statement of the problem okay now I want you to notice something and with from Oz theorem right from a serum said that if a is an integer piece of prime p doesn't divide a then a to the p minus one is congruent to one mod p congruent to one mod p this problem though is not you don't end up getting a one on the on the right side it's minus one on the right side so don't just go on autopilot just always throw a one over on the you have to look at the problem make sure that you understand what it's saying okay so 17 dividing 11 to the 104 plus one translates into this right and vice versa so all right well what is this really saying so I mean there are a couple ways I think that you could go about this in a reasonable way this is the same as 11 to the 104 being congruent to 16 I'm just sort of jotting down the main ideas here I'm not trying to make this a polished solution mod 17 right so this this is what you want this is what you must prove right everybody with me here okay 11 to the 104 congruent to 16 mod 17 is minus one is 16 mod 17 so don't hurt the baby so you want to use from Oz theorem here in some way and so that what you do is you just do it the most natural way here and you notice that 17 and 11 are both primes and 17 does not divide 11 so by from Oz theorem 11 to the 17 minus one is congruent to one mod 17 right and again like I said I want to read it right here I'm just jotting down notes really okay so from Oz theorem that is of course 11 to the 16th is congruent to one mod 17 so we see what we want to prove up above 11 to the 104 congruent to 16 mod 17 well of course we're not going to be able to get to 16 by raising both sides to powers because one to any power is still going to be one right so the point is we should be close and then whatever else we need let's see if we can just verify that independently and then just put it all together at the end okay and I'm not going to do this entire thing but you've seen this trick before of course so you want to get to the 104th power right so let's see if we can figure out what power to raise this to okay so what do we want okay well okay yeah so the point is why did I choose six because that gives us 96 if I use anything bigger than that we're going to overshoot the 104th power so once we've got this this is what we get right you might say well okay well that has taken us closer but we still have a little bit more work to do right so now the question is okay what is it that we really want additionally we want think about this we want to get to 11 to the 104th power so what we really would like here is 11 to the 8th to be congruent to 16 this is where you have to you do have to keep in mind some of the theorems that we've established for congruences and this is probably where I'm going to stop here but I just want to make sure that you can see what I'm saying I'm also not saying this is the only way to do the problem this is just one way that I think is reasonable if you can establish this then there's a theorem that goes way back that says if a is congruent to b mod n and c is congruent to d mod n then ac is congruent to bd mod n I believe that's in the book I think it's definitely true let's see yeah it's in the book theorem from 4.2 okay so you can see how this finishes the problem right because you can then you can just multiply them together and then you get what you need okay so it makes sense so this is the other thing that you should be looking at and yeah this other part I should also say that you can do this without using you might think I need to use from us theorem or something you really don't need to to do that here and in fact I don't even know if that's going to apply in this case but I mean what you can what you can do I'm not you know I'm not I'm not even gonna say anything else it's you don't have to do much I'll just say that but for this for establishing this if you want to do it this way don't be thinking about from us theorem to do this okay you just yeah I give it away if I say anything else I'm not going to but think about it okay okay so I wanted to talk about this just because of the minus one here but okay so this one so everybody have this down now okay so number two I'm actually gonna let you think about for a bit I will probably tell you something on Thursday I will tell you that to be you're going to essentially what you're what this is going to amount to is you're going to apply from us theorem a couple of times and then you're going to have to show that eight divides a to the six minus one and so this is something that you might have to resort to some earlier techniques to do this I don't see a real slick way of doing this off the top of my head I look at it for a minute I'll think about a little bit more but from us theorem is really not going to apply directly to showing that he divides a to the six minus one it is something certainly you can do it using the division algorithm and using the fact that a and 42 are relatively prime but requires a little bit of work but I'll see if I can give you a hint if I see a shorter way to do that but thinking about the division algorithm is something that will you should be able to get it at the very least yes no it is but what I'm saying is you're going to take your the natural way to do this is to kind of take them separately since it's relatively prime so then you get it at the end so you have three cases you got you want to take care of three and seven and eight and then you then you get it for free by what we've done before okay now yeah sorry I didn't I wasn't clear on that part so the three and the seven are pretty straightforward I think they work out more or less really quickly just using for mass theorem at the age is the one where you some of these things are not going to it's not going to be that from us theorem is sort of a silver bullet that just takes care of everything for you immediately you might you're going to have to work a little bit with some of the other ones yes but that doesn't really help you because to yeah I mean if you know I mean for example if you know that two divides a number and three divides a number that you can get six dividing the number but if you if you know that two divides a number and two divides a number of course you automatically get that but that doesn't mean that four divides it because two divides two and two divides two before doesn't divide two so you can't do that unless they're relatively prime in general yeah so I think one thing that you can do you still might have to do a little bit of work but there are ways that you can sort of break this down right eight of the six minus one you can actually factor okay and you also know so this is what I'll say for now you can factor eight of the six minus one and you also know that maybe I should write this down sorry I mean lazy okay so two yeah sorry about that all right week off of work and get lazy okay so really that what you're what you're trying to show this is this is just part of the problem you have to do that eight divides a to the sixth minus one okay all right so I'm just giving you a couple of ideas I still a little bit behind so I didn't I didn't really get to look at this as much as I wanted to but eight of the six minus one is the same thing as a cubed squared right minus one or one squared difference of squares which is a cubed plus one times a cubed minus one should remember this from basic algebra calculus now this is something that I will tell you that you may not have known but this you can look this up online if you want to so remind you of this maybe some of you didn't know this but you can't when I say can't I mean in the normal way using just real numbers you can't factor things of the form x squared plus a squared can't factor those but you can factor things of the form x cubed plus a cubed these can be factored and so this is the sum of cubes this is the difference of cubes because one and one cubed are the same thing now you're going to get actually you're going to be able to break this down into four pieces the product of four things okay and if you mess with it I haven't actually this I just kind of did this off top my head if you mess with it you might be able to just get the get the result to come out right away but there's one thing you should notice let me just write this down now again when I'm saying this factor I don't necessarily mean that this is going to work or that this is all that you have to do that's not what I'm saying but this is the kind of stuff you should be thinking about because if you if you want a to divide this thing well it's not enough just to know that two divides it that doesn't give it to you you can say that remember the GCD of what it was it a and 42 is equal to one right this is given in the problem and that what does that tell us so a has to be odd right now you might say okay well great if a is odd then powers of an odd number odd so a cube plus one then should be even a cube minus one should be even that takes care of four but that's not quite enough we really need to eight to pop out somehow okay so I would encourage you to play with this a little bit and see what you can come up with okay so I didn't really say too much about about this let's see here I'm going to say something about number let's see what number is this so five two yeah six so I'm going to talk about six a here I will probably give you a hint for the other one next time I'm not going to do the whole problem for you but six a find the unit's digit of three to the one hundredth I don't think that I didn't look through the section that carefully but I don't think that the book the author gave you an example of this in the in the section so some of you might be thinking how do you how do you do that how do you really convert this to a congruence okay using okay and I should also sorry sorry this is really crooked here but the direction specifically say to do this using from us there that's what it says so if you just do this the way I always say I gotta come up with some other lines but if you do this the way you would have when you were in eighth grade you could probably still solve this problem pretty quickly actually this is not that hard even if you didn't know anything because you're going to see a pattern three of the first the unit's digit the ones digit in other words is three three squared it's nine three cubes 27 multiply by three it's going to be one right then you multiply by three is going to be three and then you're going to see this it's just going to cycle through it's going to keep cycling and the pattern is going to repeat and you'll be able to see what the unit's digit of three to the one hundred excuse me three to the one hundredth is very quickly without applying from us there but that's not what the author wants to do and I don't want you to do that either I would like you to see how to apply the theory in the section to do the problem so how do you do that well how do we translate the ones digit or the unit's digit into some sort of a congruent type problem well let's just look at a specific example unit that can't talk today the unit's digit of say 54 right is what you all know what this is for and what's the relationship here well also note that 54 is congruent to four I'm running out of room here mod 10 right okay so here's the idea and I think if you just try to listen and think about it I don't think writing anything down is really going to help you here whatever the the ones digit of a number is it's that is the same thing as what the number is congruent to mod 10 that's always the case and the reason is because if you just take that ones digit off and make it a zero that's zero mod 10 because 10 is going to divide it because it ends in a zero so then when you tack it back on that's what you get mod 10 right that makes sense sort of okay so what you're doing if you want to find the unit's digit or the ones digit of a number is you just want to see what it's equal what is congruent to modulo 10 that's all you're doing here so I'll tell you that this problem is there's not a whole lot going on here I will tell you can I go to the next page here now yeah okay all right so let me change this to a W okay I'll just write this out more formally now we want to find the integer a which satisfies zero less than or equal to a less than I was right at this way with three to the one hundredth congruent to a mod 10 right that's just another way of saying we want to see what is congruent to my debt okay so how can you do this well and I will say something about this because I don't know that we did this in lecture but you can immediately apply for my theorem or the corollary to this because 10 is not a prime right that only works for primes I did give you a corollary though that says that basically if you can break the number up in the product of two primes and you can just kind of put the pieces together roughly I think I gave that to you I should have so three squared is congruent to three mod two this is from us there well actually it's not it's the corollary to from us there which says that for any integer a in any prime p a to the p is congruent to a mod p and so there's a can be anything it could be divisible by p even that was the corollary like I said so since two is a prime any number squared is congruent to itself mod two okay and three to the fifth is congruent to three mod five right and that's also the corollary eight of the p is congruent to a mod p and these are both primes so we're good here now what you can do is put these together I will actually use the number in the book here but if I didn't give this to you somebody should yell at me here but I'm pretty sure that I did four three five two okay this was well you know where well where I'm going with this is it lemma on page 89 lemma on page 89 it's just called it lemma I didn't see there was any number in here okay good all right I'm not totally worthless let's see okay so what that says then is that you can if the primes are distinct the modulus is the both primes in the modular distinct primes then what what this lemma says and this should be in your notes is that you can just multiply the powers together and so in this case we get three to the tenth is congruent to three mod ten right make sense let me see I don't think so so this lemma says that oh right right right right okay so the other the but actually let me see so right I don't think let's see if this is actually going to matter though okay so that yeah so this this lemma I didn't look at this closely enough here so the lemma says that you need a to the p right a to the p is congruent to a mod q and a to the q is congruent to a mod p okay so yeah and so in this case I guess that's probably that's not true in this case is it let's see here okay so yeah yeah yeah yeah okay so this is not scratch that this is not what we this is not what we want here but let's see here but let's see all right so this let me sorry about that yeah no no no this is not what we want to do okay scratch that whole thing okay okay so let's see this lemma you can probably let's see here so hang on yeah I just briefly skim this okay yeah so this is not okay this I don't think this lemma is going to apply directly here because even trying to flip it around is just then it's just not true what you need in order to apply the lemma so I'll tell you what let me let me come back to this let me come back to this problem on Thursday so yeah sorry about that well I had to screw up at least one problem this this semester I think this is the only one I actually screwed up okay so too late too bad all right um here let's like I said let me let me come back to this on Thursday yes I think this is the last one I'm going to say anything about 8 of the 9th is congruent and I'm not going to say much about this one 8 of the 9th is congruent to a mod 30 okay so what I will say here this is all I'm going to say for now but is simply to note that this is really of course you already know this but this is 2 times 3 times 5 right okay and you let's see let me make sure that I don't tell you something that's not true okay yeah all I'm going to say here is so think about this and this the fact that we can break this up into primes and if you put these pieces together you can actually you can get this to work out there's one little trick involved here but I don't want to say anything about that right now I might give you another additional hint on Thursday but this one is fairly straightforward okay so let's see all right I guess we have a little bit more time to kill here anything um one would like to ask about yes okay okay okay so yeah say that what it was again I had what did I have something to go into any of the 12th congruent 25 yeah it's because it's because think about translated into a divisibility problem a to the 12th congruent to one month seven means seven divides a to the 12th minus one a to the 12th congruent to one month five means five divides a to the 12th minus one because seven and five are relatively prime their product divides a to the 12th minus one okay you can always yeah you can flip congruences um there's a theorem that kind of collects all of these in chapter four are you talking about his question now or something different smush things if okay if you have the um here's what you need okay I mean you can do this in general here's what you really need though I'm just going to use a and b here so suppose you have a congruent to be um so for the one or two online students I finally learned my lesson at the end of the semester question is when you have a bunch of congruences when can you sort of smush all the module I together to get one big number well what you need here is for the two guys on the left and right to be the same so for example if you have something like this I'm just using a and b but if you have something like this if and one and two I'm just doing this for three but it works in general are um what are called pair wise relatively prime then is congruent to be mod the product and again this just really is just a translation of divisibility Euclid's limit type result from chapter four or chapter three I don't maybe no chapter two I guess was it yeah chapter two so the point is if you have both of these being the same then what this is really saying is remember this is just the same thing as the assertion that and one divides a minus b so and one divides a minus b and two divides a minus b and three divides a minus b so then the product divides it because they're relatively prime makes sense okay two and three um no uh in general you can't do that because then what you're saying for example just make to make it simple suppose you have a congruent to be mod n one and a congruent to be mod n one while you can't say that a congruent to be mod n one squared for example necessarily because you know um I mean you could take a to be four b to be two and and one to be two for example and then yeah it's not going to work no yeah you can't go up with powers here um you you really need all of them to be to not share any common factors okay any other questions you want to ask we'll do we'll talk a little more depth on Thursday about about some of these no no okay sounds good um like I said I'm going to get the webpage updated and um solutions will be posted to the exam pretty soon so be looking over that to help you