Hi everyone, my name is Raheep Prakash. And welcome to the first class of Reminders. Okay.
Reminders is a very important topic from your exam point of view. In every exam, you'll see around one question of Reminders for sure, right? But I don't know Reminders is the most hyped and talked about topic of quant, right?
One of the most hyped and talked about topics in quant, Reminders. You'll see many questions on Reminders, many tough questions, many baseless questions also I have seen, right? And too many practice on Reminders I have seen. People, my students doing.
Too many practice, too many questions on Reminders, right? Many people enjoy it, many people do it out of frustration, I don't know. Okay, so you get one question from Reminders, that's fine.
But you don't need to spend so much time on Reminders. Reminders, one question is fine and you should get it in exam, right? So, I prepared this 8 to 10 videos.
I'll be preparing this 8 to 10 videos in Reminders for you, right? You'll not get a single question or single concept outside these videos in any of your examination, right? Especially for CAT examination, I will be discussing many unknown concepts like reverse Euler theorem and all which will make your remainder solving very easy.
So this kind of all the kinds of concept especially for CAT, Chinese remainder theorem and Euler theorem, reverse Euler theorem all will discuss in remainder. These 8 to 10 videos you watch it, you won't get any question outside this in examinations. So let's start remainders. There are two special properties of remainders. First one.
The denominator, right? I'm just like writing in plain language, right? No technical terms and nothing.
Okay. Just what do we speak in those terms? Denominator can be used multiple times, can be used multiple times. Fine.
So let me give you an example of this. If I, if I write this, let's say 72 by 10, what is the remainder? So you say, okay, 72 by 10, what is the remainder? Reminder is 2. That's for sure, right?
2. 10x7 is 70 and remainder is 2. Right? So I hope you remember this that in long division method. Okay suppose I write like this.
Okay let's take example here. Right? Suppose I want to divide this 8 with 1, 2, 3, 4, 5, 6, 7. Right?
So what you do? You keep on dividing. Right?
You divide 8x1 is 8. So 12 minus 8 is 4. Again 3 comes here. 8x5 is 40. 40 again subtracted 3 comes here. Okay then again you take the next number 4. And you keep on doing this, right? You keep on doing this here till you get a number, okay? That is, this number you get, that should be less than 8, okay?
That number you get, that should be less than 8, right? This is the remainder. This is the remainder, right?
So, when long division, when you're continuously dividing, right? When do you stop? You stop only when you get a number here that is less than 8. That is less than divisor, right?
So, here this 8 is called divisor, okay? This 8 is called divisor. Okay?
This whole is the number and this is the quotient. This is the quotient. Right? Fine. And this is the remainder.
This is the remainder. Okay? So you can see here that I divided, I used 8 here only one time. Right?
Because what is this quotient? This quotient basically, this quotient could be written as 1, 2, 3. Okay? This quotient could be written as 1, 2, 3. 4, 5, 6, 7 divided by 8. What is the remainder?
And this is the process to solve it, right? So, you can see there's only one 8 here, but this 8 I'm using multiple times. 8 I'm using dividing once, again and dividing again by 8, again and dividing by 8, again and dividing by 8. So, we keep on dividing by 8. We keep on dividing by denominator till we get a number that is less than 8. That is the remainder, right? So, that's what I have written here.
I can use denominator. I can use the denominator multiple times, right? I can use it multiple times upon my, it's my wish, right?
8, this denominator, how many times I use, it's my wish, right? So, how that makes us easy here, you see? How that make it easy? Suppose I write here a number 132 by 10, okay? What is the denominator?
132 by 10, what is the denominator? Okay, we can do it easily. 10 into 13 is 130 and denominator left is 2, fine.
But if this 132. I split like 11 into 12. 11 into 12 by 10 remainder. Okay, 11 into 12 by 10 remainder, right? Now, since I can use this 10, 10 is a denominator here, right? 10 is a denominator.
Since I can use this 10 multiple times, so I can do it, I can divide like 11 by 10 remainder is 1. 12 by 10 remainder is 2. Alright, I told I can use the denominator multiple times. Okay, same. This is because of this property of long division method.
Okay, so I am using this 10 two times here. Okay, this 132 is splitted as 11 into 12 by 10. So, 11 by 10 remainder is 1, 12 by 10 remainder is 2. 1 into 2 becomes 2. So, 2 is the answer. Do we got here?
Same thing, same answer. That means basically what? If we divide the big number like this, Then also we'll get the same answer.
And if I split the big number and divide like this, then also we'll get the same answer. Right, then also I'll get the same answer. Right, I hope it is clear. I hope it is clear.
Okay, now, now I'll show you this one. See, suppose I take another example. Suppose I take another example, like suppose number is here 504. Okay, so 504 by 5 remainder.
What is the number? Very easy. Answer is 4 because 5 into 100 is 500. So, what is the remainder?
Reminder is 4 here. Reminder is 4. Fine. Now, if I split 5084 as 7 x 8 x 9. Correct. 504 can be written as 7 x 8 x 9. I'm just showing this splitting that this property of remainder is that denominator can be used multiple times. So, very useful property.
Okay. 7 x 8 x 9 divided by 5 remainder. This I want.
Right? So, I can now, since I can use it multiple times, I'll do it individually. 7 by 5 remainder is 2. 8 by 5 remainder is 3. 9 by 5 remainder is 4. 2 into 3 into 4 is 24. Now 24 by 5. So I told you right, 24 you can't write 24 as remainder right.
2 into 3 into 4 you got 24 you can't write 24 as the remainder because remainder has to be less than divisor. This remainder has to be less than 8. What is the 8 here? 8 is divisor.
So remainder has to be remainder has to be Less than divisor. It has to be less than 8 for sure. Okay, that means 24 cannot be the answer.
So what can be the answer? So what you do in long division method? Once you get a number that is greater than 8, you again divide by 8. Once you get a number greater than 8, you again divide by 8. So once you get a number greater than 5 here, that is a denominator.
You again divide by 5, again take the remainder. 24 by 5, what is the remainder? The remainder is 4. So 4 is the answer.
4 is the answer. right? You got it right? I can split it.
I can split it very important property, right? So what I'm telling you is say it's not taking any number right? Suppose I take 17 into 18 into 22 into 23 divided by divided by 15. What is the remainder?
So I don't need to multiply this big number, right? I can simply do it individually because this denominator can be used multiple times. We have seen last two examples on this on this 17 by 15. What is the remainder? 18 by 15. What is the remainder? 3. 22 by 15, what is the remainder?
7. 23 by 15, what is the remainder? 8. Okay. So, 2 into 3, 6. 6 into 7, 42. 42 into 8. Again, don't multiply, right? Again, don't multiply completely, right?
So, what we'll do here? 2 into 3, 6. 6 into 7, 42. So, I'll write 42 into 8 again, right? It will become very easy once we introduce the concept of negative remainder, right?
Just hold on, okay? Just hold your seat belt and understand this concept. Very, very important.
I'll come to negative reminders and we can do it very short. I know we can do it very short, in very shorter way, right? So, right now, simply concentrate on this funda. So, 42 by 15. What is the remainder here?
Right? What we got is, we got 42 into 8 from here, right? So, again, I know that I got a number that is greater than 15. So, again, I need to divide by 15. That's it.
15, now, again, do it individually. For 15 to the 30, I got 12 here, remainder. And 8 by 15 remainder is 8 only.
8 by 15 remainder is 8 only. Until we don't introduce a negative remainder concept, remainder is 8 only. Okay.
So, 8 by 15 remainder is 8 only, right? Because 8 won't divide 15. So, 8 is the only option left. So, 8 is the number here.
12 into 8, 96. 96 by 15, what is the remainder? So, 96 by 15, remainder is what? Remeinder is 6. So, 15, 6, 90. and remainder is 6. So, we got a number that is less than 15. So, for sure now this is the remainder.
This is the remainder. If this is less than 15, this is the remainder. Therefore, what is the remainder?
Remeinder is 6. Remeinder is 6, right? Very important concept, okay? That splitting.
So, what is the first fundamental remainder? Splitting. We can split a big number and when dividing by a denominator, that denominator can be used multiple times, okay? This is the first point concept, right?
Now come to second one and then we can combine those two. Okay, second one very important concept of negative reminders. Concept of negative reminders. Okay. Now see what is, what are negative reminders?
What are, what are negative reminders? Right. Understand this point here, negative reminders.
Okay. Okay. So negative reminders. Okay.
What are negative reminders? See, negative reminders is a concept, is a concept which we use for our own convenience, right? Our own convenience.
You'll nothing find like negative reminders or any option ever as in exam. You won't find any option ever as negative reminders. Okay. It's a concept which we use for our own convenience and we always return the answer to positive reminder.
Okay. We always, always return the answer to positive reminder. Always return the answer to positive reminder.
Right. I'll show you how. Okay. See, I told you if it is like 8 by 10 reminder, what is reminder 8 by 10? So, 8 by 10 reminder is 8 only.
Right. Because 8 will not divide 10, so whole of 8 is the remainder. Right?
Like 13 by 12, what is remainder? So, 13 by 12 remainder is what? 1. Because 12 will divide 13 one time.
So, 12 one here 12 and remainder is 1. Since 13 is here. Now, if I say that, if I say that, okay 8 by 10 remainder is 8. If I can write that, if 8 by 10 remainder is 8 is fine. It is a positive remainder 8. What is the negative remainder? What is the corresponding negative remainder for this positive remainder 8. So, corresponding negative remainder for this positive remainder 8 is what? Minus 2, minus 2. This is the negative remainder, okay?
Minus 2. Because 8 by 10, 8 is 2 less than 10, okay? So, what is the corresponding remainder here? Minus 2 for plus 8. So, I can write remainder as minus 2 or plus 8, right?
Plus 8 is the, plus 8 is the positive remainder, plus 8 is the positive remainder. and minus 2 is the negative remainder. So, every number, every number which you divide by the other number will have a positive remainder and the corresponding negative remainder.
We have to use in a smart way to save time. Okay, to save time, right? We have to use in a smart way to save time, right? So, how to save time basically? Basically, it means that in remainders, there are many powers question.
A numbers exponential question, right? So, in those questions, what we'll do? We'll use the It doesn't matter, right? We use negative or positive remainder.
What we'll do is we'll use the smaller magnitude. Because if it's plus 8 and minus 2, I'll use minus 2 for question because it has a smaller magnitude. Here the magnitude is 2 only. Here what is the magnitude?
8. Magnitude means number without sign. Okay. So here the magnitude is 8. Here the magnitude is what? 2. So we'll use minus 2, right? Now, see, like 12 by 13, 13 by 12 is 1. If I change it to, what is the remainder of 12 by 13?
Reminder. So that means 12 will not divide by 13. So whole of 12 is the remainder. So but plus 12 is the positive remainder What is the negative remainder?
So negative remainder is what? 12 is 1 less than 13. So minus 1. Okay minus 1 So what does the remainder here? 12 by 13, what is the remainder? Negative remainder is minus 1 and positive remainder is plus 12. Correct.
Okay. Now if I say here, okay suppose 37 by 8. What is the remainder? So what you can do here is 8 to the power of 32. So remainder is 5. So 37 by 8, 8 to the power of 32 is 5. So plus 5 is the positive remainder. What is the negative remainder?
So since for negative remainder what you can do is this plus 5 you can again divide by 8. Okay. So what is 5 by 8 remainder? So 5 by 8 remainder is minus 3. Do it mentally right.
Don't write here. Do it mentally. This plus 5 what you can do is this plus 5 you can just Divide mentally by 8. Okay. So, what is 5 by 8 remainder?
Minus 3. 37 by 8. What is the positive remainder? Plus 5. What is the negative remainder? This positive remainder again divided by denominator.
So, 5 by 8 is? What is the remainder? Minus 3. You get minus 3. Okay.
5 by 8, what is the remainder? Minus 3. So, very important, right? Very important.
Okay, one more example I'll do. Right? See, suppose I write here, 58 take something anything, right? Take 58 by 7. 58 by 7, what is the positive remainder? 58 by 7, what is the positive remainder?
7, 8 is 56, so remainder is 2. So, plus 2, I can write. Okay, or what is the negative remainder? For negative remainder, just divide this 2 by 7. Mentally, again divide by 7. Okay, what is 2 by 7?
What is the remainder? Minus 5, what is the negative remainder? Minus 5, because 2 is 5 less than 7, so minus 5. 2 is 5 less than 7. So, plus 2 or minus 5. So 58 by 7, what does it mean?
Plus 2 or minus 5. Okay. plus 2 or minus 5. Okay, I hope it is clear this part. Yeah, one more we can do to give you more clarity.
133 by 16. What is the remainder? So again, what is the positive remainder? So you can see 16, 7 times 1, 1, 2. 16, 8 times 1, 28. 16, 8 times 1, 28. So what is the positive remainder? 16, 8 times 1, 28. What is the positive remainder?
Plus 5, plus 5. Okay, or What is the negative remainder? For negative what we will do? Divide this plus 5 by 16. In mind.
So 5 is how much less than 16? 11 less. So plus 5 or minus 11. Plus 5 or minus 11. So 133 by 16. 16 8 is 128. So remainder is plus 5. And or what is the negative remainder?
Minus 11. One more thing I can negative remainder by. Just taking the next multiple. Like 16 8 is 128. So 16 8 is 128. So, 133 is plus 5. Difference of plus 5, right?
So, positive remainder is plus 5. For negative remainder, just go to next multiple. 16, 9 is 144. Now, 133 is how much less than 144? 11 less. So, minus 11. Just another way of thinking, right?
Nothing else. Do it everywhere, right? You can do it everywhere here also.
Like 7, 8 is 56. So, remainder is plus 2. Take next multiple. 7, 9 is 63. 58 is 5 less than 63. So, minus 5. Correct. Here also. 8, 4 is 32. Reminder is plus 5. 8, 4 is 32. Reminder is plus 5. Okay. Now, take next multiple.
8, 5 is 40. 8, 5 is 40. Here it is 37. So, how much less? 3 less. So, minus 3. Correct. This is the concept of negative remainder.
Right. So, now we will combine these two concepts of positive remainders and negative remainder. And then. And with that concept also, that divisor can be used multiple times.
Combining these concepts, we can create very good questions that we'll see now. Okay. And that has been asked in so many examinations, right? Now see, this is a question you suppose.
18 into 19 into 20 into 21 divided by 17. What is the remainder? What is the remainder? 18 into 19 into 20 into 21. Divided by 17, what is remainder? I just divided it individually.
I can do it now. Okay. So, 18 by 17, remainder is 1. 19 by 17, remainder is 2. 20 by 17, remainder is 3. 21 by 17, remainder is 4. 2 into 3 is 6. 6 into 4 is 24. I got a number here 24, which is more than 17. So, I should again divide by 17 because remainder is always less than divisor. Divisor is what?
Denominator. What is the denominator here? 17. So, 24 by 17. What is the denominator?
7. 7 is the answer. Correct? Now, what I'll do?
See, for negative reminders, I'll take one concept question now. Okay? Now, see, 14 into 15 into 16 into 17 into 18. This divided by 19. What is the remainder? Now, see, what I'll do here?
14 by 19. What is the remainder? So, in positive remainder, it is plus 5 plus 14. Okay, you can understand this thing here. 14 by 19 remainder is plus 14. Okay, in positive remainder because 14 will not divide 19. So, whole of 14 is the remainder.
14 by 19, 14 is the remainder. Or, but what is the negative remainder? So, for negative remainder, again what we will do?
14 is how much? 14 is 5 less than 19. So, minus 5. Minus 5, right? 14 or minus 5. So, since minus 5 has the smaller magnitude, that is 5. Here the magnitude is 5. Here the magnitude is 14. So I'll use 5. I'll not use 14, right? Because both are same numbers to me.
Plus 14 is a positive remainder and its corresponding negative remainder is what? Minus 5. So both are same thing for me. So what I'll do? I'll use this minus 5 here because of lesser magnitude.
Okay, easy to calculate. So 14 by 19, remainder is what? I'll use minus 5, not plus 14, minus 5. Same thing, 15 by 19, positive remainder is plus 15, negative remainder is minus 4. 15 is 4 less than 19. So minus 4. So I'll use minus 4 because of the smaller magnitude.
16 by 19, minus 3. 17 by 19, minus 2. 18 by 19, minus 1. Correct. Minus 5 into minus 4. 20 into minus 3. Minus 60 into minus 2, minus 120. Number I got is minus 120. Just look at the magnitude. Don't look at the sign.
Just look at the magnitude. Magnitude is 120. It is more than 19. So again, divide by 19. Okay. Now 19 6x1 is 4. So 6 is the remainder.
So minus with minus sign and 6 is the remainder. 120 by 19, 19 6x1 is 4. So 6 is the remainder, right? Minus 6. But, but I told you that we'll never ever write the negative remainder as the final answer.
And you'll never get in any options also in the examination. Okay. Because negative remainder is a concept which we use for our own convenience. It is not as any particular number or any actual concept. Just for our convenience concept.
Okay. So, what is the remainder of minus 6 now? So, see. See for example here.
We have just discussed this. 10 by 12, what is the remainder? I told you, 10 by 12, positive remainder is plus 10. What is the negative remainder? Minus 2. Okay, so you see here, if you get 10 by 12, if you get negative remainder is minus 2. So, how to get this positive remainder? 12 minus 2 will give me 10. Okay?
Denominator added with negative sign of remainder, 12 minus 2 will give me 10. Right? So, this will get the positive remainder. Okay? One more example. See, 13 by 17. What is the remainder?
Positive remainder is what? Plus 13. What is the negative remainder? Minus 4 because 13 is 4 less than 17. Okay. So, it is plus 13 or minus 4. So, plus 13 or minus 4. So, if I get remainder is minus 4, how to get, if I suppose I don't know the positive remainder here.
Can I get it? I can get it very easily because plus it is, the denominator is plus 17. Plus 17 minus 4. will give me 13. So 13 is the answer, right? So same thing here also, I got a negative remainder here, but I always should mark the positive remainder as my final answer. So here, what is the denominator?
19. 19, what is the negative remainder? Minus 6. So 19 minus 6 is equal to 13. So 13 is the answer, right? That basically means that if I multiply, if you multiply it, 14 into 15 into 16 into 17 into 18, you'll get a big number, right? If you divide that by 19 conventionally, you will get what is our deal get 13 as the answer only nothing else, right? So this is much easier way of doing it, right?
So get a negative remainder and always return to positive element Just by adding to denominator adding to denominator with sign 19 minus 6 is equal to 13, right? I hope you got here right now Let's do one cat question this question came in cat one once right easy only old cat question. Just I can do it for practice 1421 into 1423 into 1425 this divided by 12 a previous year cat question right can do it easily right uh what we need to do don't we don't need to multiply just uh we can see divide one number like 12 to do it mentally now so 12 1j 12 okay 12 1j 12 so 2 is left 2 and 2 22 again 12 1j 12 10 is left 10 and 1, 1, 0, 1. 12, 8, and 96. So, 5 is the remainder.
5 is the remainder, right? Now, I don't need to again divide 14, 23. Don't waste time here because 14, 23 is 2 more than 14, 21. 14, 23 is 14, 23. See here, what I'm saying is 14, 23 is 2 more than 14, 21. So, obviously, I'll get the number here that is 2 more than this. That is 7, directly. 1423 by 12 what i'll get 7 because till 21 remainder is 5 you add 2 you get this number so 5 plus 2 is what 7 okay same thing again till 12 by 1423 is uh 7 remainder is 7 and again 1425 is 2 more than this right so again i'll get what what is the remainder i'll get i'll get that 9 2 more than 7 is 9 okay this is difference 1423 to 1425 again difference of plus 2 so again Add simply 2 here. So, 5 into 7 into 9. Okay, again your wish, right?
What you can do is do it more combiningly, do it individually, right? So, you can do it like this also. 5 into 7 into 9. What is 5 into 7 into 9? So, 5 into 63. 7 into 9 is 63. 5 into 63 by 12 remainder.
Okay, 12 by 5 just 60, remainder is 3. 5 into 3, 15. 15 is again more than 12. So, again divide by 12. 15 by 12, what is the remainder? Remeinder is 3. So 3 is the answer for this question. 3 is the answer for this question.
Right? So there's multiple ways of doing it. Just need to use this property that this denominator can be used multiple times. Or what, or right, or what you can do.
I think this can, this would have come in your mind also. Right? That if it is 1421 into 1423 into 1425 divided by, divided by 12. Okay, divided by 12. So here, if I got as 5. So here, I am getting 7. Here, remember I am getting 7. But 7 by 12, it is minus 5. That is a smaller magnitude.
So yes, we can use it, no issue. I can write minus 5 also here. Instead of 7, write minus 5. There is absolutely no issue in this.
So, minus 5. Again, I know that here remainder I should get is 9. Okay. So, what is 9? So, 9 is 9 by 12 is minus 3. Should write here minus 3. Yes, write minus 3. No issue. Again, what do you get?
5 into 15. So, you get 5 into 15 is 75. 5 minus 5 into minus 3 is plus 15. Plus 15 into 5 is 75. 75 by 12. What is the remainder? 12 is 6 just 72. 3 is the remainder. 3 is the answer.
I got 3 only here. Right? You can just play with the numbers, right?
Once the concept is clear, negative remainder, positive remainder, denominator can be used multiple times. You can just play with this concept here and that's what we'll do in whole number system, okay? Numbers, everything, right?
In the remainders concept, so many remainders concept, then with applied with, applied with, usability rules, right? All we can just play with the numbers. If we have, if we are very clear with this negative remainder concept, okay?
So, answer is 3 in this case, right? Answer is 3 in this case. Okay?
Now let's do a few example here. Now suppose the number is 38 raised to 138 divided by 39. What is the remainder? Okay, so nothing to worry about. This is absolutely same thing, right? Because again, 38, how 38 raised to 138 means what?
38 into 38 into 38. This can be written 138 times. Okay divided by divided by 39 remainder. Okay, so what we'll do again same thing You will do divide individually 38 by 39 remainder is minus 1 38 by 39 remainder is minus 1 38 by 39 remainder is minus 1 right minus 1 how many times 138 times so minus 1 raised to 138 Okay, so minus 1 negative sign raised to a even number sorry Negative sign raised to an even number.
That means positive. Right. So, it is plus 1. So, since we get a positive remainder as the answer, answer is plus 1 only. No issue. Answer is plus 1. That's the answer.
Right. Okay. Now, see. Now, had the question been here, 38 raised to 137 by 39. Same thing.
So, now you can do directly. 38 by 39, remainder is minus 1. Minus 1 raised to how many times? 137 times. Since... my negative sign raised to an odd power.
So that negative sign, it will retain. So what is my answer? Answer is minus 1. Answer is minus 1, right? 1 power anything will be 1 only. Minus 1 power anything will be 1 or minus 1, right?
See the concept here. The concept is 1 raised to x is equal to always 1. x can be anything, right? Minus 1 raised to an even power will always be 1. Minus 1. raised to an odd power will always be minus 1. Because minus 1 into minus 1 into minus 1, 3 times, right?
It is minus 1. Because for even number of times, it is always plus 1, no? Minus into minus plus. For even number of times, always plus. So, that extra in odd number 1, extra minus is there. That plus into minus becomes minus.
That's why, right? So, minus 1 raised to 137 is minus 1. Odd power, so minus 1. Okay. But now, minus 1 cannot be my answer. Because minus 1. is negative remainder. My answer should be always positive remainder.
My final answer should always be positive remainder, right? So, how to convert to positive remainder? I told you.
Just add to base with sign. So, 39 minus 1. So, 39 minus 1 is what? 38. So, 38 is the answer.
38 is the answer for this question. Right? You got the difference here, right?
It was even power. It is odd power here. Correct? Okay. Now, let's do some more.
Like see here, 3 raised to 196 divided by, let's say 4. What is the remainder here? Let's do it easily. 3 by 4 remainder is minus 1. Minus 1 raised to 196. So, power is even. So, what is the answer?
Answer is 1. It is positive remainder, so 1 is the answer. That's it. We don't need to do anything here, right?
Otherwise, 3 raised to 87 divided by 4. What is the remainder here then? 3 by 4 remainder is minus 1. Minus 1 raised to 87. It is odd power. So, minus sign it retains.
So, minus 1. But my final answer is never negative. It is always positive. Just add to base with sign for the positive remainder, for the corresponding positive remainder.
So, 4 minus 1, it will give me 3. So, 3 is the answer here. That's it. 3 is the answer.
Okay. So, I hope this basic part is clear to you. Okay. So, in next video.
We'll be doing some good questions on it also right and different applications. Okay, thank you.