like what does this mean when we have a is congruent to b mod n and all the following are pretty much equivalent and depending on the situations you can interpret them differently okay so perhaps the first thing i want to tell you guys that when we have a is congruent to b mod n this means that a and b have the same remainder and i will put on quotation mark for the word remainder because sometimes when we're talking about this kind of things you may end up with negative remainders and things like that so that's pretty much ideas that's why i put on quotation marks a and b have the same remainder when [Music] they r divided by n okay and by the way n should be a positive whole number greater than one so two three four five and so on do not ever say n is equal to 1 mod 1 kills everything don't do that okay you're pretty much just killing all the math killing all the numbers you eat so don't do that okay so this is the first way to interpret this notation a and b have the same remainder when we divide a by n when we divide b by n so that's the first thing the second thing is that i can write the following when we have a is congruent to b n we can say a is equal to well it's just going to be off by a multiple of n and usually we can just say k times n for some number k which we don't know yet just keep it as how it is for now and then you add b to it and you see when you mod n this right here states as a when you have k times n ma n this is always going to be a multiple n so it becomes zero in the world and then you're left with the b right here okay so sometimes it may be helpful when you go from here to here because from here you're working with an equation rather than a congruence so that's another way to use this notation i will say another way to interpret another way to approach when you have a congress the next thing is that okay i can just subtract b on both sides right and if you subtract b on both sides you can see that a minus b is equal to k times n in other words a minus b is a multiple of n right so i can actually go from here and tell you guys that and divides into a minus b and once again this right here has a few ways to interpret it this is actually the common notation that we'll be using quite often this right here you say that's device okay and when we write this down this means uh let me just put this down in blue like why not this means that a minus b is a multiple of n okay so that's the idea so that's pretty much it right i think this is just the perhaps the most natural way to know what that means and this right here is pretty useful when you're trying to solve one equation um from congruence right you change the congruence to an equation first and then this right here is useful when you're doing some proofs and perhaps i will just give you guys a quick example okay so this is just a quick example so let's do a easy one i'll say let's say we have 10 okay and that's congruent to 14 mod 4 is it yes it is why because when you have 10 divided by 4 this right here gives you 2 with the remainder 2 right so this right here is pretty much 2. and then right here when you do 14 divided by 4 you get 3 with the remainder 2 as well so they have the same remainder and once again sometimes you may be dealing with negative number and i can also tell you guys that 10 is congruent to negative 2. mod four okay and this is two on the left hand side this is negative two but the truth is that you can just go from negative two and you add four to it negative two plus four this is also the same as two mod four okay so that's the idea now the second thing is perhaps that is trickier because i told you guys we have this huh and i want to write this down okay a is my 10 right here based on the order and this is the first way the second thing is that i will write 10 is equal to k i don't know yet multiply by 4 and then we add b which is 14. well ten is equal to what times four plus fourteen this is fourteen i just need to have negative one so that i can get negative four plus fourteen which is ten so k in this case will be negative 1 k and a and b they are all integers and the third way as i said n divides into a minus b so n is 4 divides into a minus b which is 10 minus 14 in another what we're saying that 4 divides into negative 4 why because we know negative 4 is equal to 4 times negative 1. this is a multiple okay and once again we are allowing negative numbers like this right so hopefully this right here makes everything clear and from here i feel much better because now i can just tell you guys about new things in my number three videos and stay tuned let me know if you guys have any questions or any comments or any suggestions and at the moment that's it