Guys, welcome to another webinar session. I am happy to start with you today. And before we start with number theory, I have a warm up question for you. Some of our friends already entered, I mean, joined our session earlier. If we get some answers about some of the answer choices, I can go from there. And start solving it. Okay more and more friends are joining. Let me give you like 45 seconds. Please share your answer directly to Soham guys, so we can just move quickly So for four people answer a Okay one friend says a and my end to it was Five look like a lot of friends says it should be a it has been 45 seconds. Anyway Let's see if the answer is actually a Progress for this type of questions. You do not necessarily do any calculations Why I am saying I'm gonna explain that so because as you have only Use eight five or three here. Look at the answer choice before you do any calculations guys and We either add some of them here as you see or just left over the another number To be able to see which fraction has the biggest value We need to have the largest value at the top you know largest as a numerator and smallest number In the denominator if you can get both of these conditions your number is going to be largest value as you see among three five and eight guys eight and five are the biggest numbers the smart move should be Adding those two numbers at the top and left over the smallest number at the bottom in that case answer would be a and if you think about the smallest value you would say three, you know the smallest value at the top and Some of the largest values at the bottom that it would give us least value Why I said you do not necessarily find answer guys, let's see They kind of play around and give us like huge numbers they use 130 135 and 140 Imagine if it was the case I mean, I mean even I made it, you know easy to add If you had given those three four whatever digit numbers you would still Try to find out Where they add two of the largest numbers If addition and you know, if anything is only the case then you need to find the smallest number as a denominator Remember, you do not need to add them or you do not need to calculate them for these cases All right. I hope that makes sense any question for me here If you have you can just ask us via chat really quick or I'm gonna move on I Believe you're good here All right Since we have new friends are joining almost every week. Let's just start with I'm gonna continue with the problem solver strategy Make sure guys, you know what they are asking. I Would just underline especially The you know question mark part the last sentences Sometimes they want us to find the total but you find only one of the variables even you need to find both of them At the end, please be careful with that Plan how to solve the problem. Are you gonna use algebraic ways? Are you gonna create equations? Are you gonna use some of the table graph picture? Whatever if it is geometry It is better to you know, divide those complex shapes two into two or three different parts or some of the cases from geometry as I'm saying If they give you half of the figure, you know, semi circle half of the triangle whatever You can complete them to the full Circle or full solid shape that you can work with that Carry out your plan and carefully complete your calculations and organize your thoughts and steps Make sure Your calculations are correct. You do not have to rush to guys if you have like some answers in your mind Anyway, in many cases when you read the question or see the problem The first thing first solution method you think is going to be Right or it's close to right one, but it still may not right after you just check and plug in to answer Choice to you know, check it if it's correct So what I'm saying you just gotta be careful With your calculations look back and check and reflect whatever answer you find make sure it kind of makes sense you know, we have been told that a lot if they are asking you to find distance, it cannot be negative or some other common sense forms And I'm gonna move on We're gonna talk about number theory here Let's just check some of the topics really quickly Set of the numbers the numbers we used to work math problems can be categorized into sets Look at for these terms in problem statements Common sense includes because I mentioned counting numbers Remember, they are the whole number starting one. They are also of course positive numbers Remember, we cannot start counting by zero. That's the reason the big difference between counting numbers and whole numbers is being zero here Integers we still have whole numbers But when we talk about integers, we add also negative whole numbers that full set gives us integers rational numbers Every single repeating this repeating terminating decimal or any type of you know, fractions are Decimals, I mean those are rational numbers if we have non-perfect squares or pi e those type of special cases we call them irrational numbers so when we just show them in a Set let's just say this counting numbers this is whole numbers and this is rational numbers case As we can say, of course every counting numbers are rational. Yes, that's correct Every whole numbers are also rational. Yeah, that's correct. But some rational numbers cannot be either whole or count numbers this value in the decimal system of numbers each digit to the left as a Value ten times greater than digital. It's right. For example, one two, three four is ten times greater than one two, three point four Also ten thousand is greater than nine nine nine nine thousand nine nine nine point nine nine and because it has a digit in the ten thousands place as you see You are gonna look for after that decimal point. We have four digits and we have five digits here Even that's number on the left seems bigger. It's not the case some Rules of the number theory I Didn't I didn't end zero properties. They have us simplified problems For example, if you add any number or subtract with zero you get same number if you multiply any number by one you get the same number If you divide any number by itself you get one Of course as long as the number is not zero because it's a different case Zero put the property of multiplication means when you multiply a number by zero you get zero When you divide zero by any number you get zero But when you divide a number by zero, it is undefined. So you cannot find an answer with that Negatives are opposites. For example, negative numbers is opposite of their positive can sometimes help you calculations have with calculations When you add any number with its negative You got you gotta get zero Now you multiply any number by negative one you get negative number which additive inverse of that number and May you multiply negative one by negative number you get the positive product Or I will maybe you remember that case where you multiply two negatives it become positive They used to say it's gonna give us kind of smiling face, you know It might be helpful as well for you to Even an odd number even numbers can be divided into two equal parts. They are multiple of you know To open recognizing if number are even or odd and how they interact with none of your answer choices For example, when you add odd and odd number you get even all the time Even plus even you always get even odd and even get the odd If you do not know how to memorize you can say the when you add same type of number You Always get even you know If you get different if you add different type of numbers, you always get old When you multiply all you get old Besides that condition every other choice gives you even number as a result when you multiply them To be able to find odd number when you multiply you have to have both of numbers are all numbers Other multiples closer other common multiples in your problem solving as well. Remembering the some rules of divisibility Also can save us time in problem solving If you know, how does divisibility of 3, 4, 5, 6 works? You can tackle the problem easily All right You Powers or exponents represents repeated multiplication For example, you know instead of multiplying a times a times a like three times. We just say a cube These are the representing same thing The most common powers are squares and cubes Obvious x squaring is x times x, you know x cube means x times x times x of course x square literally represents area of square x cube is Representing volume of a cube which side length of x Roots are kind of undoing of powers square root of 100 gives us 10 because 10 square was 100 you'll know that When you have cubic root of x cube That degree of power and the power itself cancel each other if they are the same that you have left over x Another way to write Root is to use fractional power For example when we have 4 here in the I mean 4th power of 4th root of 16. We gotta keep the base and Remember every single number has invisible power here, which is 1 The power goes at the top and degree of the root goes bottom in the fraction guys As you see we can just proof Those cubes and cube root cancel each other from previous example x cube in the cubic root can be written as x to the 3 divided by 3 Which is x to the 1 power is equal to x. I Assume if you have any questions, you would ask us guys I might go a little fast to be able to you know, solve more and more problem together But please yeah, you know, you can directly chat to me or you can chat you can send message so If you get stuck or if you see something is not clear Of the roots we have some of the exponential rules we can use and make it faster for us all problems Exponents represent a repeated multiplication. You know that several shortcuts make calculations faster. We suggest you prove these Suggest you prove this to yourself product rule means guys We have a to the x power times a to the y power at the end You should have a to the x plus y what I meant again, if it doesn't make sense Let's just you know assign them Some numbers. Let's say we have 2 to the 4th power times 2 to the 5th power Equals we both know that 2 to the 4th power is 1 2 3 4 times 2 you multiply by each other and to the 5th power is 1 2 3 4 5 2 so you multiply each other at the end you have 2 to the 9th power if you just count those twos or At the end you see a 4 plus 5 is also 9 You can kind of you know prove that if it doesn't make sense to you When you divide Exponents with the same logic you are gonna do subtraction. For example, let's see I have 5 to the 7th power over 5 to the 5th power This gives us 7 5s together 1 2 3 4 5 6 7 Over we have 5 of the 5s 1 2 3 4 5 Then you just cancel them You may think these are these 5s look like s. I am pretty sure I am sure about that But they are actually 5 not s and then you get 5 to the 2nd power Because 5 to the 7 minus 5 is also gives you 5 to the 2nd power when you do the calculations All right power rule means power of power Here you gotta multiply them because that represents a to the X Times a to the X times a to the X. This is Y times. Of course, you gotta do, you know multiplication here Powerful product rule gives us you kind of distributing the exponent here guys in the every single term here One of the common mistakes. Let me show you let's say we have 2 to the X square Raised by cube Kiddos just tend to do it is 2 to the X to the 6th power And you see the problem here guys, you can apply that through the I mean that power of 3 here in the X You get X to the 6th power, but that cube also supposed to be applied to 2 as well Then you should get 2 cube or 8 to the you know 8 times X to the 6th power When you then power of a check for a fraction rule that exponent in the fraction also Apply every single term as a numerator and denominator Zero exponents any number to the raised raised by zero gives us always one Then you see negative exponents. You are going to use Inverse of product of the numbers or recall them reciprocal we had a over 1 if you just ignore the power, you know a over 1 becomes 1 over a and That's negative power that negative X exponent becomes positive X This is how you convert negative exponent to positive exponent because in many cases They are gonna ask you to write answer with the positive exponent guys not negative anymore Fraction next one. We all are talking about this one. You may see sir. I am not sure which one goes to the Top which one is bottom which one is numerator which one is denominator. We both know that Number of the number represents by power goes to the top guys Because when this number getting bigger, of course fractions also get bigger and you know, when you get like bigger Powers, that means the result is gonna get bigger Of course degree is gonna go to the denominator when that degree big when the degree gets bigger Numbers get smaller with the same logic when you work with the fractions when you compare 1 over 5 and like 1 over 9 we both know that 1 over 5 is greater than 1 over 9 because This number gets bigger that means The value of fraction gets smaller and smaller Any questions so far guys, we're about to start solving problems I Haven't seen any chat. All right, let's just start Checking some of the problems together this one from 2012 and number 6 In which of the following expressions can replace each occurrence to the number 8? by the some positive number Look like other than 8 and obtain obtain same result how it is possible You are going to change everything and make different number different same number, you know, but still you get the same result This is the tricky part Let's just start your timer You Okay, it has been one minute so far. So will we get some answers or not yet? We got 2 for E. Okay. Actually 3 for E. Okay, what about others? And then we got 1 for A and 1 for C. Alright, look at that. We have all of those different ideas, that's cool. So guys, let me show you what I meant. When you do some calculations guys, you are going to find some sort of special numbers. Either like 0, 1 or whatever. I'm gonna tell you why. Let's just check A. When you get 8 plus 8, you get 16. First of all, I can just remind you from order of operations, you gotta use PEMDAS. Remember, we gotta use parentheses first. Whatever we have in the parentheses, it comes first. Then, if there is an exponent, use it, which is not the case. Then from left to right, multiply and divide. From left to right, add or subtract. So, start with parentheses. 8 plus 8 is 16 here. Then we have division and addition only. Then we just divide. 16 divided by 8 is 2. 2 plus 8 is, I mean, 2 plus 8 is, yeah, it's 10. We got 10. 10 is not some sort of special number. I'm gonna tell you why. What about B? We gotta add those 8 and 8 first. We gotta get 16. If you have only multiplication and division, you just gotta do that. 8 times 16 is 128, I believe. Then divide by 8, again, gives us 16. C says, we have only plus and minus, add or subtract from left to right, just add and subtract them. 8 plus 8 is 16, minus 8 is 8. Another 8 is giving us 16. We have 8 plus 8 is here. 16 minus 8 is 8. 8 times 8 is 64. And from here, we have 8 plus 8 is 16, minus 8 is 8. 8 divided by 8 is 1. As I told you before, guys, you are going to find some sort of special numbers, which is 1 or 0. If you remember from the previous slide, some of the properties, they were related either 0 or 1. For example, if you multiply any number, remember, any number by 1, you always get the same number. With the same logic, if you divide any number by itself, you always get 1. Look, I am saying any number here. And when you check e, you get the result as 1. That means, instead of 8, it doesn't matter which of those four numbers you replace here, with those, you know, same operations, you will always get 1. And they said, you are going to obtain the same result, okay? That was the reason you are looking for some special case. I believe we are good. Okay, if it's not, I believe you are going to ask us via chat anyway. I am going to move on. And number 2. These type of polynomial questions might be a little tricky. Please be careful when you work with those. What's the difference between the largest 6-digit polynomial and the smallest 5-digit polynomial? Let's just start the timer. 3 people answer B. All right. Let's see if we have more. Because compared to 3, we have a lot of friends here guys. I mean, I will depend on only 3 people. Another person says B. All right. So far we have 1 minute and 20 seconds guys. Maybe we can wait for a couple of more friends. Another person says B. Look at that. They just keep saying B I believe. Another and another, you know, and another. It's about goes infinite. All right. We finally have a majority of people who say B. Okay. Let's see if it's actually. Okay. I told you here, be careful with those numbers. Like for example, when you look for the largest, you know, palindrome, if you largest 6 digit palindrome, if you think, sir, I can start with 9, you know, and make it like 987, then 789. Then I can say, but you are doing wrong because it's never says that some of the digits are going to be different, you know. So even that number is palindrome. Instead, because, you know, here is the example given because that example kind of misguide us to use different numbers in first three actually getting different. But what if I choose every single numbers as nine from those six digits. Can I do that? Yeah. This number from right to left, left, right. It's going to get the same number anyway. So that means it, it applies to room. We did number and the smallest five digit palindrome. Well, we're talking about five digits number. I live on a user, but if I just put the zero here, it's not going to be right. But remember, I cannot say the smallest palindromic, whatever, you know, five digit number one, two, three, and two, one, it's not the case because we can use some of the zeros. Even we can use bunch of zeros, you know, why not? You know, smallest five digit palindrome number should be 10,001. At the end, you got to do some, you know, subtraction. Since it's really easy, I'm not going to even, you know, waste our time. I believe when you do that, you should get answers. Any questions about this one? I believe we are good. Cool. Number three, please careful with the wordness of the problem. I mean, like exactly what they're asking is really tricky. Is not the sum of three different one digit number is. And right away, someone answers D and another person answers E. Okay, that's good. We'll start with some answers. It's cool. It has been only 10 seconds. Another person answers D. Okay. And now the E person changes to D. Oh, okay. Another D. Another D and another D. Nice. I hope you actually know the answer. You do not just guess because someone says D. If someone says answer in 10 seconds, at least they were right. I mean, it's not the case all the time, you know. Maybe our friend do like small calculation mistakes. Who knows? Just saying. It has been one minute and 20 seconds so far, six friends says. Some of the answers. Okay. Maybe we can just begin. Yeah. Start. So, okay. Yes. Here. We are looking for some of those two digit number. Okay. This is not the sum of three digits, not the sum of the three different one digit number. Look, we can go with the answer choice if you like, you know, it shouldn't take too much of time anyway. We have 10. Can I write 10 as a, you know, five plus two plus three? Yes. Since I can rewrite 10 is a sum of three different one digit number, it cannot be answer, you know. 15. I can say, let's see, seven plus six plus two should give us 15. It works. But this is which one is not, when I check 23, let's say I can say nine plus eight, another number should be is six or to get 23, yes, yeah, six, you're right. Then you see, we can find 23 as well. Well, we get only 25 and 28 guys, as you see right now, we have 50% of chance. So after that piece, before even check which one should write answer, I can say, let's find some of the biggest value of sum of three digit, sum of three different one digit integer. So we can say nine plus eight plus seven. When you add them guys, you should get 24. As you see, if we had answer choice in the 24, we would eliminate that as well, but we have two of them. We have 24 here on the number nine, then we have 25 and 28. As you see, they are saying you need to find the smallest two digit number. That means guys, you're approaching to 24, but at the same time, you're looking for the smallest number. You have either 25 or 28, we both know that 25 should be the answer. It's not still 24, it's bigger than 24 because we cannot obtain that answer when we add those three different numbers. It's not 28 either guys. That was the case, answer was 25. Any question about that? It's kind of related inequality logic problem, you know, at the same time, of course, number theory. I haven't seen any chat. That means we are good. Look at you. All right. Next one. Number four, please check number four guys and let me start timer for you. so far we had like 49 seconds so far well not too much answer for it but not yet at least one minute and eight seconds finally someone answers should be this is they say be gotcha so I believe here I'm also little sick so and now someone answers see look at that yeah B versus C what about others zero zero and zero oh my gosh three more people answer see oh boy maybe get one or more answers so that I can just start it's almost two minutes anyway and another person says see okay okay that's enough so let's just start guys look like majority of our friend says answer should be C let's see one of the kiddo was yeah anyway so wrote down several consecutive integers okay I will you remember because integers they come one after another so like you know three four five six and so which of the following could not be the percentage of all numbers among them all numbers among the means among them represents the whole and all numbers represent a part part overhaul we are going to compare guys think about this before we start to solve the question we need to check some different variations for example if I just start with even numbers so I have like you know two three four five six let's say I started even an end with even you know here as you see I am going to have two odd numbers and three even numbers okay let's see if I just you know started even and end with old two three four five six seven right now as you see guys if you just compare old number even numbers I have three odd numbers versus three even numbers okay what if I start with all numbers like one two three four five as I start with old and end with old it's gonna give us two three odd numbers versus two even numbers and what if I just start with odd number and end with even number four five six well right now I have three odd and three even numbers case oh my gosh look at my long hand writing okay three odd and three even numbers so here is a deal when you compare all numbers versus even numbers when I say compare guys for some case you either subtract or divide look at that there is always look there is always I mean either odd is equal to even as you see or the difference is always one you know it doesn't matter or odd minus even is one or they are equal to each other there is no other case guys you can check other you know scenarios it doesn't matter I kind of try to every single of the scenarios so with that in mind let's just kind of simplify those given percentages if I say 40% is equal to 4 over 10 I believe you all know that and when you divide by 4 and 10 by 2 you get 2 over 5 let's check 2 over 5 first and they say we have all among them if I say 2 odd numbers and 5 is total that 5 can be written at 2 odd and 3 even remember the difference between odd and even number is just one which was our case because it is possible as you see here what about 45 percent 45 percent means 45 or 100 they divide both sides by 5 you should get 9 over 20 I believe you should get 9 over 20 guys well let's say we have 9 odd numbers okay here what we would get here 9 odd and 11 even numbers hmm it's gonna be possible let's say because I remember many friends says for date let's just check for dates also 48% means guys for they divide by 100 you know when I divide both sides by 4 I should get 12 over 25 I believe 12 over 25 guys well they say odd numbers among them odd numbers over all of them we can separate that 12 I mean 25 by 12 odd and 13 even guys all right what about 50 this is about as you see 50% half of half that means odd is equal even you know I mean 1 over 2 or odd is equal even it's it's obvious we all know that and what about 60 60% means 6 over 10 or 3 over 5 that means we have 3 odd we can make it 3 odd numbers versus 2 even numbers it's even you know one of those answers I mean one of the examples I just showed you after that answer guys after the explanation which one do you think is the right answer can you please share your answer one more time if you really get that explanation now again I am saying answer may or may not be seen I will have one friend share their answer with me please guys do not share answer with me I can check it but it might be a little late one friend will give us one answer I wonder about others what they think finally people are getting the correct answer what they say finally two people say B all right one of friends also says B that means we get three votes I'm not sure about others guys again think about it look this one mild question I mean middle middle one number 15 it's not that too hard or not too easy or whatever but please be careful guys when you check those as you see there is some sort of rules between the number the possible number of even and odd number in one of those consecutive integer sets as you see there cannot be difference of two here where you check 11 and 9 guys it's not possible that was the reason answer was B all right any question for this one since we get like majority of this one wrong I am asking this any question about this one or can I move I really wanna move by the way okay we are moving first clear the board then get out of here okay number five this one was one of my favorite I wonder how many of you are going to try to find 2013 to the sixth power whoever get first they get little gifts let's see which of our current find 2013 six bar first or do we really need to find that value that's the question you ask yourself yes this one was number 22 one of the hardest questions but still are they gonna get hard since you cannot use calculator in the exam anyway let's start timer Bye bye. One person says E. Look at that, okay. And another person says D. Alright, that's good. We have different answers. And then the E person changes to C. Look at that! And one person answers B. Another person answers C. What was it? Another person answers C. And now the B person changes to D. And then another person says D as well. And then we got one C person changed to D. Oh, it looks like so far many of our friends said D. Let's see what's gonna happen. Alright guys, I'm not sure why it has been more than 2 minutes. Guys, I don't know what was your thinking process. But of course I don't want you to think of 6% of 2013. Instead guys, can we check 2 to the 6th power? Not just 2 to the 6th power. 2003 to the 6th power. And let's try to find numbers of squares from 1 to the 2 to the 6th power. And also we gotta find cubes from 1 to the 2 to the 6th power. 1, 2 to the 6th. 2 to the 6th power is equal to 64. Thanks for the hint, I will know that. Let's S be the number of squares among those. Okay, I am finding S. S is what? How many square numbers among 1 through 64? Let's just count. We have 1. I believe you all know that. Then 4, 9, 16, 25, 36, 49 and 64. As you see. That amount of numbers. We have 1, 2, 3, 4, 5, 6, 7, 8 numbers. 8 numbers here guys. What about Q? Number of cubes? Let's just find. We have 1 is a cube of 1. Then 2 cubes basically give us 8. Then 3 cubes is 27. Then 4 cubes is 64. As you see guys, at the end, we just get what? 4 numbers. So from there, what would you do here guys? How can you get some sort of generalization here? Let's say S is equal to Q. Is it possible? Not necessarily. Because we get 8 numbers and 4 numbers. Compare those. They say 2 times S is equal to 3 times Q. Even if you multiply them, it's not. Because guys, remember, to prove something is wrong, you can just use only one statement. And that works actually. If you say 3S is equal to 2Q, then you multiply them. 8 times 3 is 24. 4 times 2 is 8. It's not even close. Let's check E. Since more and more friends say D, let's just check E. It says S cube is equal to Q squared. Well, what's S cube? S cube is 8. 8 cube is 64 times 8, whatever it is. And Q squared, we get 4 times 4. It's not even related. It has to be this one. So you are going to say, but sir, it's not going to work. S is equal to 2013 times Q. I am aware of that because I only give you this one as an example. Because we cannot find 2013 to the 6th power. It's not possible, guys, logically. But do you know how we get that 2013 times Q? That part, I kind of wonder. Remember, in that number of patterns, we used the last number was 2013. In our pattern, the last number was 2. Since you keep thinking, I'm going to show you that. In our pattern, we start with 1 and we end with 2 to the 6th power, you know. According to this one, it says S is equal to 2013 times Q. That means S is 8. If I say 8 is, guys, 2 times 4, which is correct, I can prove my point. Because remember, I didn't use the same number pattern. I used something similar. The only thing, I mean, I had the same rule. Everything was the same. I only made the last number 2, as you see here. That means it works because we literally still have the same pattern, same rule here, guys. I still have the number to the 6th power. I didn't change anything. As you see, instead of 2013, if my last number is 2, then that should work. Because it was 2 actually here. I don't know if you have any question or not. I might get a message if our friends ask something. I don't know. Okay, answer was D guys, majority of our friends get it right and they get it correct. Any question about this one? I know this one was a little off, that's the reason from either last week or a couple of weeks ago. I told you, if you have given like big numbers, make it smaller. Of course, do not work with 2,000 terms, it does not make sense. We all know that. Instead, use 2 to the 6, even if you can use 3 to the 6 power, you know, 3 to the 6 power is not. 27 times 27, you can manage somehow, you know. 27 times 27 is what's that, 24 times 3, 900 something, I guess, I don't know, whatever. Okay, I haven't get any more chat, that means we are good, maybe we can solve one more or more questions, that's it, all right. Number 6 guys, let's check this one. We have 30 stories, okay, it started on a new page, obvious, the length of stories are in that order, on the first page, what's the largest number of stories that can start all numbered page, interesting, remember, we discussed the relation between odd and even numbers, I hope you remember that. Let's start your timer, guys, okay, and answer E, okay, one person says E, one person says E, that's cool, okay. Remember guys, you need to find that the largest number of stories that can start on odd number, how can we make that? You know, we can play around with the order of those number of, order of different stories which has different number of pages, so. Guys, I shouldn't get only one answer, I mean, come on. At least do some educational or whatever, yes. Or like non-educational, literal yes, you know. One person says A. Okay, good, you see? You get different answers right now. Gosh. And now answer E. Another answer, I believe, for E. Okay. If I get just one more answer, I'm gonna start, come on. I mean, compared to four answers, we have a lot of friends here. Others, are you sleeping or what? Okay. Another person answers A. A, all right. We have two A's versus two E's. Let's see, guys, what's gonna happen. Maybe answer is nine. All right, guys. As you know, our goal is to get largest number of stories which can start an odd-numbered page. Well, we both know that the first story can start page number one. You know, that's the first case. Let's say, let's just think about this. If I start with odd-numbered story, like from page number, let's say page number start with the one-paged story. So one page is just gone, then we just continue with page number two. That means we gotta start with even number. Nope, we don't want that. But they say largest number of stories that can start an odd-numbered page. What if first story is even-numbered story, like two pages or 30 pages or whatever? Let's say first story has two pages and it started with page number one and one and two is gone from first story. Then next story has to, you know, start from three, which is an odd-number. Well, well, well. Maybe, guys, since we already started with odd-number, we are going to use every single of 15 even pages. Remember, one through 30, starting two, four, six, and so far, we have 15 even numbers, okay? Even number of pages. If I use them, I have to have 15 odd-numbered page already. Then I have only odd-numbered page left. I left over one, three, you know, five, seven, and 29. So far. So when we start story number 16, guys, 16 is also going to start with odd-number. Remember, because of those rules, when you just fill in the blanks about even-numbered page, it start with odd-number and end with even, then another story has to start with odd-number anyway. So, you know, we just show you. Start with odd-number, end with even, which is one and two. Then has to be start, second one has to start with odd-number, which is our target, you know, value. So that means after of those 15 cases, 15 even stories we use, guys, number 16 story also start with odd-number. But since that story had odd amount of pages, well, even just think about, imagine from here. You start from number one, if you had only like three pages, one, two, three, end with odd, then start with even, guys. Okay? The next one, after one, 17th story is going to be start with even page, which is not work. But since this one has odd number of pages, because we have only odd number of pages left over, that one start with the last page of that 17th should be even number. And next one is going to be still odd number, 18th. 17th doesn't work, 18th work. And so far with that logic, guys, 19th shouldn't work, but 20th should work. What I am trying to tell you guys, after 15 and one more, we had 16 story used, we had total 30 stories, 30 minus 16, we had 14 story left to fill in. But among those 14 stories, guys, as you see, only half of them are going to work our condition, you know, because they kind of switch places. Well, in that case, we can work with 14 over two, which is seven more stories are going to start an odd number page, you can test them if you like. I mean, I already showed you how I test that. Well, we had 16 plus seven gives us 23, guys. Whoever says answer is E, you get it correct, man. I am proud of you, look at that. Look, this one was number 26, was one of the tough questions and you get it right. I mean, I am proud of every single of you, but I am more proud of those whoever says E. Any question for me today? Okay, that means no, I guess. All right, guys, that means you get it. I am proud of myself too, you know. So I explained well, I guess. All right, guys, thanks so much. And this is the end of our session. And I am going to see you next time, all right? Bye.

number theory

interactive exercises

problem-solving

mathematical laws

divisibility rules

strategic thinking

collaborative learning

critical thinking

palindromes

properties of exponents