Use Modular Arithmetic to Solve Equation | ISI BStat BMath Entrance Obj 17 | Olympiad Number Theory

hello everyone I hope you are doing beautiful mathematics in this particular video we will learn a technique in Elementary number theory that is useful for mathematical Olympians like iqm American math competition this particular problem is from isi Bad bmath entrance we will learn a lot using this one problem I will also give you a challenge problem at the end of this video give that a try and if you can put in the comment section so this problem says that X and Y are integers and x * y * x + y + 1 is equal to 5 ^ 2018 + 1 we want to find out how many solutions are there of this equation how many solutions are there of this equation we are going to use something called modular arithmetic modular arithmetic it's a very powerful tool if you do not know modu arithmetic check the links in the description you will find many interesting videos and explanations of modular arithmetic in short it is a way to combine remainders and do calculations with remainders and uh if you can do it then a lot of complicated number Theory problems become much easier so let's look at this the right hand side we will do a modular check of four okay so mod four check why are we doing this mod for check there are variety of reasons it comes with experience really but if you look five five is obviously congruent to 1 mod 4 so 5 to the power to whatever is 1 to the power that thing mod four and that's just one mod 4 so it is natural to try and experiment with modu 4 because the right hand side becomes much more simplified really the options are modulo 3 modulo 4 modul 5 these are the obvious choices because if you do that then the right hand side becomes much more simpler so we will do it with module 4 okay so let's write this down carefully XY * x + y + 1 is equal to 5 ^ 2018 + 1 the module of four the right hand side is congruent to 2 mod 4 because five is congruent to 1 mod 4 and you can raise both sides to the power 2018 as a I did just a moment ago so that's just one and then you can add one to both sides so you get 2 mod 4 that's why I say the right hand side is 2 mod 4 you want the left hand side to be also 2 mod 4 okay the question is is it possible can X okay x y x + y + 1 be congruent to 2 mod 4 for some values of X and Y let's check this is a hard check you have to check by numbers so X can be congruent to 0 1 2 3 y can be congruent to 0 1 2 3 we really have to check 16 values 16 pairs 0a 0 0a 1 0a 2 like this 16 pairs first of all notice that X cannot be 0 mod 4 so X can so this is not possible why because X is X cannot if x is 0 mod 4 then that the left hand side entirely becomes 0 mod 4 0 * anything is 0o but the right hand side is 2 mod 4 right so the left hand side is 0 mod 4 or divisible by four the right hand side is non divisible by four so that's not possible so we cannot have we cannot have 0 mod 4 so zero and this also cannot be zero y cannot be zero now we can check with one so basically we can check 1A 1 1A 2 1A 3 these are the three pairs we can check this x this is y this is X this is y this is X this is y so let's check with 1 1 * 1 1 + 1 + 1 that is congruent to 3 mod 4 doesn't work because it has to be equal to 2 mod 4 right that's what we want then let's try 1 comma 2 1 * 2 1 + 2 + 1 this is congruent to 0 mod 4 why because this is four right four is 0o so four is 0 mod 4 so the left hand side is 0 mod 4 so again doesn't work because we need it to be 2 mod 4 so this one doesn't work this one doesn't work 1 comma 3 1 * 3 1 + 3 + 1 this is congruent to 15 which is congruent to 3 mod 4 again doesn't work has to be 2 mod 4 that's what we want so none of these three pairs none of these three pairs work so now you can check with 2 comma 2 2 comma 3 2 comma 1 if you want you really don't have to check 2 comma 1 because youve already checked 1 comma 2 and two and one are interchangeable the left hand side X and Y are interchangeable so you have to just check 2 comma 2 2 comma 3 and 3 comma 3 three more things can you check all of these variants and tell me which one works or maybe none of them work then there is no solution so check and tell me if there are any solutions to this particular equation thank you for watching this video I hope you learned how to use modular arithmetic to find out integer solutions to equations these are called daptin equations daptin equations are equations with integer Solutions and uh we cleverly used number theory cleverly used modular arithmetic to actually solve an equation you just have to check three more cases and tell me in the comment section if there are any solution or not I just told you the strategy I hope you are liking these sort of videos and you are learning something from it if you are please sub subscribe to our Channel and if you're interested in outstanding programs in mathematical olympiads research leadership programs that lead up to IV League Ed Ed ation then check the link at the description for a mentoring program thank you I'll see you in the next one bye