CSAT: Remainder Theorem
We will discuss two very basic concepts. We can call it division and multiplication and questions based on that, all questions based on that. So this is is a typical division. When you divide A by B, A by B where the answer is C and the remainder is R. So all possible questions based on this. Again as I have discussed in the previous numbers session also the most difficult part when it comes to preparing with numbers is knowing what to prepare or how much to prepare. It’s very difficult to predict. We will also look at questions based on product of two numbers equal to some other number. And questions which are related to numbers and its basic patterns. First we will start with questions on division. Questions on division.
Now what are the basic types you can expect where a division is involved? So the easiest can be questions based on remainders but not necessarily the most important one. Questions based on basic remainders. So first we will see what is basic remainders, or, Basic Remainder Theorem. See, we will look at questions which are of very high level of difficulty as part of today’s session, but I will simplify it to a level which anyone can understand. So what is basic remainder, if you ask me? See I am discussing questions of this category. A by B equal to C where the answer is C and the remainder is R. Now if we look at all they have two papers and questions on numbers. There can be questions based on this C. C i am calling it as the quotient, or the answer. Then there can be questions based on remainders. There are papers where you will see, or books where you will see good number of questions on remainders. How to find out a remainder – when you divide a number by another number. So, like, possible remainder questions you need to prepare. Questions based on C we also need to prepare. There is a good chance that you can get more questions on this also. So we will look at both and then before looking at questions based on products. Now products can be products of 2 numbers, 3 numbers, 4 numbers like that. These two we’ll take care of and what are the different ways of expecting questions so there might be 5, 6 different types. So today’s session is going to be very intensive but very important also. So, 3 and a half to 4 hours. First half I will discuss questions on division. Second half I will discuss questions on multiplication. So, you will clearly get an idea with all types of questions.
So basic remainder – first I am discussing questions on R. Where you will be asked to find out R. Then we will look at questions on C. And there can be questions on B also. See this is the basic division, A by B equal to C where the remainder is R. This is quotient and this is remainder. Simple example I have already written here. This is a general case. An example can be 10 by 7, quotient is 1, remainder is 3. Now, some of the remainder questions, and I am only discussing basic to moderate questions to start off with. Basic remainder is nothing but, while trying to find out remainder, if the numerator – this is a numerator. Assume that the numerator is already factorized as 37 into 38. I am giving you a sample question here. By 12. When we are trying to find the remainder, you just need to, what’s the basic remainder theorem? It’s very basic as the name suggests – you just take the product of individual remainders. Now 37 by 12 remainder is 1. 37 by 12, I am not talking about the answer here, I am talking about the remainder. So, 37 by 12 remainder is 1. 38 by 12 remainder is 2. So the answer will depend on 1 into 2, which is 2 itself. So what is basic remainder theorem, if you want to write in words? Product of individual remainders. When? When is it used, that’s important. When the numerator is factorized. So basic remainder theorem is nothing but product of individual remainders. So most of you will be knowing this, others can note down – product of individual remainders. Now if the numerator is written as a sum, that time you will take sum of individual remainders. If it’s written as a product, you will take product of individual remainders, if it’s written as a sum you will take sum of individual remainders. So that’s basic remainder theorem. And when is it used? That’s very important. That’s used when the numerator is factorized. If the denominator is factorized, as 12, if you are factorizing it as 3 into 4 you cannot use this method. No it’s a wrong method. So and why is it used if you ask me? To simplify remainder questions and to get the answer faster. So basic remainder theorem is product of individual remainders. One more sample question and these are questions of the level of difficulty which you can anyway expect. Direct or indirect. There’s one more question. So, see, this is to find out the remainder that’s why I am writing R over there, not the quotient. And numerator is already factorized in such a way you can easily find individual remainders. So individual remainders are 1, 3 here and 5 here. I am only talking about the remainder not the quotient. Quotient is 30 anyway. 360 by 12, quotient is 30. But remainders, first we are looking at questions based on remainders then we can look at questions based on quotients. Then we can look at questions based on divisors. If we can take care of all 3, it’s done. So here, answer is based on 15. Answer can’t be 15 right? When you divide a number by 12 possible remainders are 0 to 11. So 15 by 12 we will get the answer as 3. Now these are very basic questions but good chance that if you are getting remainder question, or question on remainders it can be only of this level of difficulty in your paper. Now continuing with these basic remainders now what are the other questions where you can use this particular logic of factorizing the numerator. Now basic remainder theorem is based on factorizing the numerator. So if you get a question like this 2 raised to let’s say 32, divided by, you can see these kinds of question in all the preparation books. 2 raised to 32, divided by 15. Find the remainder. So, this, I am not writing in words here. You might get the question in words. Find the remainder when 2 raised to 32 is divided by 15. One more basic question only because here also you can apply basic remainder theorem. You need to factorize, what is the common sense or logic used, factorize the numerator in such a way that, factorize the numerator in such a way that, you look for a power of 2 manually. Manually look for a power of 2 which when divided by 15 where you can preferably get a remainder of plus one or maybe minus one or zero. So that, now what’s the idea behind this? Why do we need to look for something like this? So that you don’t need to do any multiplication and here you can use trial and errors. But there will be questions where this trial and error method won’t work. So manually if you verify, look for a power of 2. See this is what you will try finding out. Look for a power of 2 which when divided by 15 where you can get a remainder of plus one or minus one. So that’s 3 raised to the power 4 right? Because we know that is 16 by 15, what is the remainder? 16 by 15, remaining part will be 1. Quotient is also 1, remainder is also 1. So what is 16? 16 is 2 raised to 4. Now. Now why we are doing like this? To factorize the numerator so I can write it as, numerator can be written as 2 raised to 4, into, 2 raised to 4 how many times? 8 times. Which in turn can be represented as 2 raised to 4, whole raised to 8 also. So, 2 raised to 4, whole raised to 8 by 15. So I am starting with basic to moderate. We will also look at higher level of difficulty questions also. Now what we are using is nothing but basic remainder theorem. So what are the individual remainders? 2 raised to 4, by 15, the remainder is 1. Now how many times you will get 1? 8 times. So product of 1 8 times and answer won’t change. It will remain 1. Now what is, what is the logic behind looking for a remainder of 1? It’s because of this so that you don’t need to multiply. So that logic is used as a calculation technique here. Looking for a power of 2, which when divided by 15 where the remainder is plus 1 was just to simplify the calculation process. Method is still the same. Product of individual remainders. Now since you are considering a product of 1, 1, 1.. eight times, there is nothing to multiply, the answer is 1. So just one more just for practice. 2 raised to 32, by 17. Here also when you are trying to find the remainder factorize the numerator in such a way that, factorize the numerator in such a way that you are looking for a remainder of plus 1 or minus 1 when divided by 17. Here again we know that 2 raised to 4, this is just trial and error method, but there are methods where you can find using some shortcuts you can get this number fast. But manually taking 2 raised to 4, by 17, that is 16 when you divide by 17. What is the remainder? Remainder is 16 itself. Now getting a remainder of 16 while dividing with 17 can also be written as minus 1. If you are getting 16 as a remainder when you are dividing a number by 17 you can also take it as minus 1. A remainder of 15 can also be taken as minus 2. Remainder of 14 can be taken as a remainder of minus 3. So 16 or minus 1 and that’s what we want. Why? Because now we can use basic remainder theorem. Minus 1, how many times? Because this 2 raised to 32 is nothing but 2 raised to 4 whole raised to 8. So we will get minus 1, minus 1 how many times? Even number of times. So answer will remain plus one. Minus one into minus one into minus one eight time. So answer will change to plus one. So, 2 raised to 32 by 17. Answer is 1. Now these are typical remainder questions where we are using basic remainder theorem. Now basic remainder theorem is nothing but product or sum, you can even take it that way. Product or sum of individual remainders based on how the numerator is expressed. If it is expressed as a product of numbers, you will take product of individual remainders. If it’s expressed as sum of numbers, you will take sum of individual remainders. So the advantage with these kind of questions is they are very easy to identify and they are sure shot answers once you now the approach. If it is see, you are getting, if you are getting the answer as minus one, you can write the answer as 16 also, as discussed before. When you are dividing a number by 17, you getting a remainder of 16 can also be taken as minus one. When you are dividing a number by 11, assume that you are getting a remainder of 10. You can also take it as minus one. If it’s 9 you can also take it as minus two. Just for representation. So that’s about basic remainder theorem.