My darlings, greetings! We've reached the second day of the 70-day TYT Mathematics camp. It was important to get through the first day, to start the camp like this, to make an introduction to the camp. Now, the second day... From now on, it will flow like water, friends. A little bit is gone, a lot remains. One day gone, 69 days left. You've achieved a lot, haven't you? Your morale has improved! Now, friends, today we will be studying the greatest and smallest values. How did we suddenly become so serious? Now, today I will explain the greatest and smallest values. You will watch two lesson videos. Right now, open your 70-day TYT Mathematics Camp Video textbook and look at the camp program section. Now, we've opened the camp program, I found the second day, the greatest and smallest values ​​video. So, where is this video? Pages 12 and 13. We will do pages 12 and 13 from the video textbook. So, what is the homework? I'm looking at the section opposite: Basic Definition Test 1, pages 12 and 13. You will also come from the question bank and solve the greatest and smallest value basic definition test 1 on pages 12 and 13. The video assignment will be finished, let's talk about the next lesson and its assignment in the next video, okay? Are you all good now, in good spirits, no problems? Yes, we started strong, camps always start strong. Friends, you start with full motivation, the first day, the second day, but the distinguishing factor, the potential for you to pass your exams, starts when you're prone to giving up. You won't always be this strong, things will happen, you'll do things in your normal private life, your girlfriend will leave you, things will happen, you'll have family problems, you'll argue with your father and mother, you'll say " enough, I'm leaving home," and then you'll come back. Continuing the camp during these periods is very important. Now you'll say, "With all my worries and troubles, the camp is important," but if you come to me with that argument, friends, during your university preparation period, you can't have anything more important than studying, okay? Everything else besides studying, except for health problems – I'm putting health aside, except for a health problem that you or your family might experience, God forbid – you can't have any other job, task, or occupation besides studying. So you put everything else aside. And you're finishing this camp, okay? With these warnings, I'm starting my lesson. Let 's see, what is the greatest and smallest value? Friends, do you know what the greatest and smallest value is? The greatest and smallest value tells us: what is the greatest value of this expression, what is its maximum value, what is its minimum value, what is its maximum value? I will focus on the solution methods and question types for this question, okay? So, this is the topic I'm talking about, the greatest and smallest value. In fact, you don't see this topic much in some books and resources; they integrate it into other units, which I think is very important. Because this is where we waste the most time in the questions that ÖSYM (the Turkish university entrance exam) asks in the TYT (the Turkish university entrance exam), because we are always trying, "Is this greater than this, is this smaller than this, what is this, what is that?" and so on. Because we are constantly trying these things, we can have problems. That's why the greatest and smallest value questions are important. Now, let's start with a direct example, sir. I beg you, doesn't this have a special feature, doesn't this have a point to consider, isn't there something? Friends, this starts directly with a question about the greatest and smallest value, just like with positive and negative, in number sets. There's no way I can give you such clear information; it starts directly from here, from the first question. We're given a set A with elements 2, 3, 7, 8, and 9. The question asks what the maximum value of the expression given for the distinct elements A, B, and C is. Let me write this expression again: 5 A's - 3 B's - 4 C's. Now, I'm going to try to maximize this expression by assigning values ​​to A, B, and C. Let's think logically. A has a positive number in front of it, right? +5. So, the larger the value I give A, the larger the positive number that comes out when multiplied by 5, and this significantly increases the expression. Therefore, I'm going to try to give A the largest possible value within this set. More precisely, what's the largest possible value I can give to A? 9. I've assigned 99 to A. Now I'll come to B... Now, both C and B have minus signs in front of them, so when I write something in B, like 9k 5 45, I'm subtracting from 45. Subtracting from 45 means reducing 45. There's nothing we can do. I wish there was something with a minus sign so I could put a minus sign in B and make it a plus sign instead of subtracting from 45. Then the expression would be even bigger, but this is what we have: 2, 3, 7, 8, 9. There's nothing we can do. We even used 99, guys. It says a, b, and c are different, so we can't use 9 for b and C anymore. Now we need to subtract a number smaller than 45 so that the number remains large. So what will we subtract from 45? What will we give instead of b? 3. We'll try to keep B as small as possible. So we'll give B the smallest element we can from set a. So, how many? 2? E. This is -4C is subtraction -3B is subtraction. Which one will I give the smallest element to? That's it here. You need to pay attention to this. Look, if you give B 2 and then C the next smaller number, 3, what happens? Look, 45 - 3 x 2 = 6 - 4 x 3 = 12. When I subtract 6 from 45, it's 39. When I subtract 12 from 39, it's 29 = 27. Look, it's 27. But if you do it like this, if you give B 3 and C 2, meaning you give the smaller number to the negative number with the larger coefficient, what happens? Look, 45 3 K 3 9, so -9 is 4 times 28 -8. When you subtract 17 from 45, what do you get? 28. As you can see, a larger value came out. So what you need to do is give the smaller number to the negative number with the larger coefficient. Or rather, if you think of it without the negatives, you need to give the smaller number to the larger number so that you get something smaller. Just so you know. Instead of subtracting 12 from 3, it's more logical to subtract 8 from 4 times 2. So, what will our answer be here? It will be 28. Okay, Professor, the answer is 28. Why didn't you try 7, 88, etc. for B and C? If I gave 7 and 8 to B and C, I would subtract 21 from 7 x 3 = 20, which is 45, so I would make the number much smaller. I wouldn't use 7 and 88 for B and C at all. Okay, the point is understood, right? Okay, I'll continue with 2. So, a, b, c, and d are digits. What is the sum of a + b + c + d? Now, if you notice, what is the minimum and maximum sum of a + b + c? It doesn't say anything here, it just asks what the sum of a + b + c is. Goodness gracious! Now, the difference between two digits is 7. What could 2 be? For example, if the difference between C and D is 2, then 9A could be 7, 88 could be 6, 7 could be 5, etc. It could be 6 to 4, 5 to 3, 4 to 2, 3 to 1, 2 to 0. These are all numbers. But pay attention, for example, let's say you take 99 to 7. Now, if you take C as 9, what do you subtract from 99 to get 3? 12. Look, you prevented B from being a number. For example, if you take C as 8, then -88 goes to the other side, B becomes 11. You prevented it from being a number. If you give C 7, throw it to the other side. More precisely, if you give 7, what do you subtract from 7 to get 2? Let's do it like this: 7. You gave 7, you threw it, it became 10. B became 10, that's not right. You gave 6 to C, C becomes 6, B becomes 9. B9 goes to the other side, A becomes 13. You gave 7, that's not right. Okay, let's continue. What else can we give? 5 to 3. Now, if you give C 5, you give B 8. If you give B 8, A becomes 12. Move on to 4 to 2. Give C 4, B becomes 12. If 7 is A1, then A1 is A1. Can we give it only one value? You have to give it 2, then 5. If you give it 5, then you have to give it 9. There is no other possibility. Okay, then A9 + B5 + C2 + D = 0. 14 + 2 = 16. Agreed? Okay, this is a nice question without asking about minimum or maximum. Does ÖSYM (Turkish Higher Education Institutions Examination Center) ask such questions? Do they really ask such simple questions? Yes, they do! We will solve many past questions. Let me give you a practice: How should the numbers be chosen to find the greatest value of the product of two natural numbers whose sum is given? How should the numbers be chosen to find the greatest value of the sum of two natural numbers whose product is given? Now, from the example below, I will explain this to you, then we will fill in the blank together. Now, look, it says the product of two numbers is given, what is the maximum value of their sum? x, or natural numbers x, y. If 18, then the natural numbers whose product is 18 are 1, 18, 2, 9, 3. 6, so 6, 3, 9, 2, 18, 1, right? Are these correct? Okay, now look. In case 1, their sum is 19. In case 2, their sum is 11. In case 3, their sum is 9. In case 4, 9. In case 5, 11. In case 6, 19. What is the greatest value here? 19. So, what is the maximum sum of x + y? 19. Now, let's give the practice. If there are natural numbers whose product is given, to maximize their sum, the numbers are chosen as far apart as possible. Let me not even write far, let me write it like this: farthest, dad, farthest. What does "farthest" mean, sir? Look, 1 and 18 are far apart, 29 are closer. 3 and 6 are closer. As you can see, this is the closest case. When the numbers are closest to each other, the smallest value of their sum is found. We can expand on this practice like this: to find the smallest value of the sum of two natural numbers whose product is given, this time the numbers are chosen as close as possible to each other. I'm giving this to you so you know the practice, not so you memorize it. If I were giving it to you to memorize, I would have written it directly. Now I've come down one step. Look, this time the two numbers... The sum is given, meaning what is the sum of x and y? 23. Okay? Now let's look at natural numbers. These are 0, 23, 1, 22, 2, 21, 3, 20, and so on and so forth. You give 12 to 11, 11 to 12, 10 to 13, 9 to 14, and so on. Let's look at the multiplications. The multiplication of the first one is 0, the multiplication of the second one is 1 and 22, 22. What is this? 20. Sorry, not 21, it's 22. What is the multiplication of this? 42. What is the multiplication of this? 60. It goes on and on. What is this? 132. Here it is the same way, 132. What is this? 130. What is this? 4, 9, 36, 126. Look what's happening. It's bringing the numbers closer together. 0, 23, 1, 22, 2, 21. The multiplication is increasing. 0, 22, 42, 60, as you can see here. It reaches 132, which is the maximum, because if you think about what came before, 10 and 13, so 130, it goes up to 130, then becomes 132, then becomes 132 again, then drops back to 130. Look what happened, it reached this point, so 132 is its maximum value. So, to find the greatest value of the product of given natural numbers, the numbers are chosen as close to each other as possible, okay? Similarly, if the sum of x + y is 23, what is the minimum of their product? It's 0. How did that happen? When you choose the numbers as far apart as possible, the product becomes minimum. So you can multiply this like this: to find a dichotomy, the numbers are chosen as far apart as possible. Okay, did we understand the issue? We will come across examples related to this, x, y, and z are positive integers. Let's go, let's see, x = 12 - y, Z = 8 + y. What is the maximum product of x and Z? Okay, come here, come here, Koçero. In our area, they call it Kero, guys. They don't say it like that at all. By the way, Kero, Kero, where was that place again? They say y there, Yido side by side. What are you doing, sir? Wait a minute, wait a minute, sir. What's wrong, sir? What are you doing now? I saw this here: x and Z are given in terms of y, and there's nothing about y in the question. I look at the front of y, I look at the front of y, i. Brother, this is driving me crazy. The sum of x and Z is x + z. These guys are mortingen. When you add 12 and 88, you get 20, so from the information given, I conclude that the sum of x and Z is 20y. Okay, the sum of x and Z is 20. Look, x, y, z are positive integers. If the sum of two positive integers is 20, what is the maximum possible product of these numbers? You're going to choose the numbers closest to each other. If I take x as 10 and Z as 10, I'll have chosen them closest to each other. What will their product be? 100. Sir, how did you take x and Z as the same? Why wouldn't I? At the beginning of the question, I say x and y are different. Are you kidding me? Is this a joke? So you're doing it. Therefore, I can take it, but if the question said x, y, and z are different, then I would take 9 and 11 instead of 10 and 10. I mean, 10 and 10 are close, but it doesn't work, so I would take 9 and 11, okay? Here, when you give their sums, if the product is going to be maximum, what do we do? We choose the closest numbers. If you get confused here, whether to choose the closest or the furthest, try a couple of them. For example, try 9 and 11, 99 is smaller than 100. Try 8 and 12, 96 is getting smaller and smaller. Try 7 and 7 and 13, 91 is getting smaller and smaller, right? So if I choose the closest, it will be the largest. Okay, my darling, my sweetheart, my love, my darling, may God not separate us. That's my only wish. Now I'm continuing with 6 examples. Oh, something got in my eye! I hope it wasn't a fly or something! What got in my eye? I was going blind! When I was little, something like this happened, a bee stung my eye. "Here, sir," I said. "Why are you stinging me here, sir?" "Is that even possible? Come on, sting me here," I said, like a wasp. And we were at a picnic, people who live in Anatolia know this better. We were having picnics in the dust and dirt, you know, in Central Anatolia, there's no greenery, no greenery at all. In Sivas, where would you find greenery? You're having a picnic in the dust and dirt, it's all trouble. That plastic ball, that rush to play with things... You know, when the meat is being cooked at the picnic, the dust gets on the meat, the plastic sheeting on the wooden table, tomatoes fall on it, you eat them like that... These are troublesome things. I don't like picnics, this awful heat, the dust gets in your mouth, it's so hot. When I was little, it was always like this, you know, when I go to picnics when I'm 45, I always sit in the car and think, " Why won't I get sunburned?" I really don't like those types, but apparently that's how it is. That's why I don't like picnics, guys, I don't like them. Where did we come from? I got something in my eye and connected it to a picnic. Let x, y, and z be integers, blah blah blah. According to this, how much greater is the greatest value of the expression than its smallest value? Let's start with " Bismillah". Now look, 3 times Z - 2 times y + 1 time x, isn't that big? Okay, now let's find the greatest value. First, let me write "eb". Oh my god, let me write the greatest value. Now, for the greatest value, friends, since Z has a plus sign in front of it, I'll give it the maximum value I can. Since X also has a plus sign in front of it, I'll give X the maximum value, but I'll give Z a value greater than X. Why? Because Z has a plus sign in front of it, multiplying it by 3 makes it even bigger. Okay? Now look, the maximum value I can give Z is 6 because Z is less than 7, so the maximum is 6. The maximum value I can give x is 4. Give 5, sir. Okay, son, if I give Z 5, is there an integer between 5 and 6? Is there also an integer between y and 6? It has to be that way. That's why I can give 4, and in this case, y becomes 5. I gave 5 to x and 4 to y. Now, do you think I did the right thing? Actually, in principle, I did the right thing by giving x a large value. By the way, I wrote the opposite. It will be like this: I will give 4 to this and 5 to that. Now, did I write and do the right thing? Actually, I did the right thing, I gave x the maximum value I could, but I forgot the multiplication sign (-2) in front of y. Now, since there is a multiplication sign (-2) in front of y, the large value I gave to Y, which is 5, is one of the largest numbers here, the number right before 6. I made the expression very small here. So, instead of trying to give x a large value, I'll try to give y a small value to get rid of this -2 problem. Therefore, I will try to give x a small value, and I will try to give y a small value. The smallest value I can give to y is 2. I can't give 1. If I give 1, it will be an integer between 0 and 1. Since there is no x, nothing is left for x, and when you give Y 2, x becomes 1. I gave Y 2 here and x 1. So what happened? This is 6K - 4 + 1 = 14 + 1, how much did we get? 15. Look, in the other case, if I had given 6, 5, 4, it would have been 18 - 10 + 4 = 12, and 12 is smaller than 15, as you can see. Since I was looking for the maximum value, this would have exploded. Now I've found the largest value. Now let's find the smallest value, let's write the expression again. It won't stick to your hand, my dear. Write it down, what will happen? 3z + 2y + x. Okay, let's give a value here, let's give a value here, let's give a value here. This time, since I'm looking for the smallest value, I'll give the smallest possible value to 3Z. The +3 in front of 3Z is positive, so I'll give it the smallest possible value. What would be the smallest value I can give to Z here? It would be 3, right? Because if I give Z 3, it would be as small as possible. I would want to give Z 2. But then y becomes 1, x becomes nothing. If I give Z 1, I can't find x and y, I swear. So I'm giving 3 here. If I give Z 3, then I have to give Y 2 and x 1. I think this is good. 3K 3 9 - 4 + 1 5 + 1, so our minimum value is 6. So what does it say? The biggest value is 15, how much more is the smallest value than 6? Our correct answer is found to be 9, okay? Okay, that's great, really great, you understood, right? No problem. Look, please don't skip until you understand. I'm explaining it as slowly, permanently, and clearly as possible. Please don't skip without understanding, okay? Okay, we continue with 7. Wow, did we solve 7 questions? What a lesson, Edis! I swear, I'm not being boastful or anything, but while I'm teaching you, I swear to God, I do n't know how time has passed. It's been almost half an hour since I started teaching. How did it all happen so quickly? The teacher gives an equality for positive integers a, b, and c satisfying the condition k, c, k, and a. He asks what the minimum value of a + b + c is. So, what is the minimum value of a + b + c? I'll write it again: Dad, a x c - 2 = 18 x b h. Now, if I'm looking for the minimum value of a + b + c, then I need to find the minimum values ​​of this friend, this friend, and this friend separately, right? I need to find the smallest possible value for them. How do I find this? How do I find the smallest possible value for them? Let's start by giving B the smallest possible value. Teacher, why did you start with B? Why didn't you go to A or C and just jump straight to B? Like a snotty mess! Because, friends, look, B is the smallest number here. If I give B the smallest value, I can work with the others. If I give A the smallest value, I might have trouble with B and C. For example, if I give A 1, I can't give them anything, etc. So, I give B the smallest possible positive integer. a b c Since it's a positive integer, I can give B the smallest value of 1. Friends, the product of a and C - 2 is 18. Now, if the product of two numbers is 18, and e and A are greater than c, then this could be 18, this could be 1, this could be 9, this could be 2, this could be 6, this could be 3. I can make this smaller and C larger. For example, if I make this 3 and this 6, C becomes 8 and A remains 3. However, AC is greater than c. So we'll stay here. Now let's look at case 1. When a is 18, let me write it as a b c. When a is 18, since C is -21, C becomes 3. I write 3 instead of c. B is already 1, which is quite large, but their sum is 22. Wow! Now let's take 9 instead of a. Then B doesn't change, B is 1. If I take A9, C becomes -2, which is 2, so C becomes 4. What is their sum? 14. Let's look at this one. If I take 6 instead of A, since B is already 1, C becomes -2, and C becomes 5. From this, 11 + Since I found that the largest and smallest value I could get from 1 to 12 is 12, I can confidently say that the answer is 12. These kinds of questions are the most tricky, aren't they? You find the answer, but then you wonder if there's a smaller one. These are actually the most difficult types of questions for ÖSYM (Turkish Higher Education Institutions Examination Center). These are actually the types of questions that ÖSYM (the Turkish examination board) puts us in most often to eliminate us. In other words, ÖSYM puts us in a coma here. Even if you get this question right during the exam, you can't say, "Excuse me, I got this right, but I hope it's correct," or anything like that. You say it, but you say "I hope so," because you might have missed something. Those who haven't experienced the exam, the exam anxiety, the exam psychology, can't know. You have to experience that anxiety in the exam. You have to overcome that anxiety with self-confidence. And self-confidence means that when we solve as many of these types of questions as possible, we will have overcome the psychological side of it. Now I'll continue with 8 examples. If you'll allow me: a, b, c, and d are different counting numbers. Okay, a x c equals D, and b equals c squared. What is the minimum value of a + b + c + d? Okay, now the product of a and c equals D, which is 1. B is equal to the square of C. Now, a butterfly came in! Get out! A B The smallest possible value for C and D is 1. Since it says their sum is minimal, I need to start with the smallest value I can give. So, I'm giving C the smallest possible value, which is 1. If I give C 1, since 1 squared is 1, B will also be 1. However, these are different counting numbers, so I can't give C 1. The smallest value I can give C is 2. If I give C 2, since c squared equals 2 squared, which is 4, B will also be 4. Here, it says a x c = d. I said 2 for C, so let's say A is 1, the smallest counting number I haven't given anywhere. But this time, since 1 x 2 equals 2, D will also be 2. However, c and D must be different because a, b, and c are different counting numbers. So I can't give A 2, I can't give A 1, I can't give A 2. So, let's give C 3. If I give C 3, then B will be 3. 2 times d6 is done. Look, I gave the smallest values ​​I could give. So, A3, B4, C2, D is 6, total 10, 15. Okay, the smallest sum of values ​​we can take is 15, right? I'm moving on to question 8, look, I'm moving on. I'm continuing with 30 things and 9 examples. Look, this is good. It says that A and B are integers such that a = k = 0 and k = b. What is the greatest value of the product a x b? Now let's put these two together. Because both of them have 'a'. In the first one, 5 remains, and in the second one, B remains. B = 5 + B = E. Let's move this 30 to the other side. This becomes -30. Now look, the product of one number and another number is equal to -30. It says that A and B are integers, and even that A is negative and B is positive. Okay? Now, assign a value to A. What value should I give? For example, let's start with -1. If I give -1, I need to give 30 here. A is negative, B is positive. Remember, if I put -2 here, then -15 here, sorry, +155, if I put -3 here, then I need to put 10 here. If I put -5 here, then I need to put 6 here. -6 here, 5 here, -10 here, 3 here, -15 here, then 2 here, and -30 here. Now look, if B + 5 is 1, then B is -4, so B must be negative and positive. If B + 5 is 2, then B is -3, that's not right. If B + 5 is 3, then B is -2, that's not right. If B + 5 is 5, then B is 0, that's not right. All of these cancel out. All of them cancel out. Now we're asking what the greatest possible product of A and B is. Okay, let's do this: let's say A and B. A comes out to -1, B is 5 + B is 30, so B is 25. In the second one, a comes out to -2, B + 5 is 15, so B is 10. In the third one, a comes out to -3, B + 5 is 10, so B is 5. In the fourth one, a comes out to -5, B + 5 is 6. Here comes Jesus B 1. Friends, as you can see, the product of the first and second is -25, here -20, here -15, and it's increasing. Do you see? Because even in negative form, when you bring the numbers closer together, what happens? It increases. When we multiply these, what do we get? -5. So, since -5 is greater than the others, what is the greatest possible value of the product a x b? Is it -5? Okay, okay? Friends, let me talk to you like this from now on. Friends, I'll go, but my name will remain. Friends, let them remember me. Yes, example 9 too. So you've turned it into a mess. What does it say? 10 x y and z are positive integers. It says x + y / z is 19. According to this, what is the greatest value of this expression? Hmm, now let me talk about the denominators. Let's not confuse ourselves with x. The greatest value of A / B, my little ones. Now, for the greatest value of A / B, you will give the greatest value here and the smallest value here because if you divide a large number by a small number, you get a large number, that is, 10. Which gives a larger number: dividing by 5 or dividing by 2? Of course, dividing by 2 gives 5. Keep that in mind. But look, I want to write this as x + y / z. Or I want to think of it as x + y / z. How? What kind of idea? Okay, teacher, how is this useful for us? Look, there's x, y, and z above, right? The question asks for the greatest value of this. Think of them as two numbers, their sum is 19. Given the sum of these numbers, what do we need to do to find the greatest possible value of their product? I need to choose the numbers x, y, and z as close to each other as possible. How do I choose two numbers whose sum is 19 to be as close to each other as possible? I choose one as 9 and the other as 10. So their product is 9. Look, x is 9, y is 10, and their product is 9 x 10 = 90. The maximum value it can take is 90. Okay, dear teacher, y/z = 10. What is Y? What is Z? What does that have to do with you? It doesn't concern you. What you need is for y, b, and Z to be 10, for example. Maybe this is 10, this is 1, this is 20, this is 2, this is 40, this is 4. That doesn't concern you, the question isn't asking you that. Okay, my dear? Let's go. When given more than one equation in a practice problem, you should focus on the common unknown. In example 11, let's look at the product of A and B, what did our friend give us? 36. And what did he give us for the product of B and C? 27. He says natural numbers. Now, look, if I do this, let's say I give this 1 and this 36, their product will be 36. There's no need for that because when you give this 36, you can't find C as an integer. You can't get 27 by multiplying 36 by an integer. That's why, to make it easier and faster to assign values, as the practice problem says, you need to focus on the common unknowns. Here, the common ones are b's. You'll go with the 'b's, you'll build the game around the 'b's. So here, you'll do it like this: I need to write a 'b' that, when multiplied by an integer, gives both 36 and 27. For example, if it's B1, it gives 36, if it's B1, it gives 27. Great! We found it! But what if it's B2? No. If it's B2, you can't multiply something by 2 and get 27. What if it's B3? If it's B3, it's A1, if it's B3, it gives 27. Great! For example, 4 won't work, 5 won't work, 6 won't work, 7 won't work, 8 won't work because none of them give 27 when multiplied by 27. So you need to look at the numbers divisible by 27, you need to look at the numbers that are multiples of 27. Let's go to 99. 9 is 9. Multiplying 9 by 36 gives 36, multiplying 9 by 4 gives 36, multiplying 9 by 3 gives 27. Finally, friends, will it give 27? No, it will satisfy this part. But this part... It doesn't provide E, meaning A, B, and C can take these values ​​separately, friends. What's the sum? In the first one, it's 28, in 36 it's 28, in 58 it's 64, in the second it's 12, 12 more gives 24, and in the third it's 12. Why isn't it the same? Okay, I said it's the same, 12 plus 4 gives 16. Look, there are 4 values, okay sir? Uh... there are 3 values. I apologize, 1, 2, 3. So the correct answer will be 3. Okay sir, why did you add them up? There were already 3 possibilities, you could have just said 3. Why did you add them up to find the solution? The question isn't asking you for the results, the question isn't asking me for the results, it's asking me something important: how many different values ​​are there? If when I add this and this, they both come out the same, then I would have said two. So when it says how many different values ​​this sum takes, you need to check that it counts the same values again, not twice. Otherwise, you'll have a huge problem. I'm telling you, you'll just explode. So, shall we continue with 12? Let's go again, partners. We'll go through it, but this question has a bit more movement than the previous one, a bit of artistry. Look, what's the product of k and m, my dear? It's 15, I wrote that here. And what's the product of m and n? It's 12, I wrote that here too. So, that's it. Now, we'll go with the common ones again. That's clear, right? The m here and the m here are common, are n't they? We'll go from there. Now, the problem here is, my family was very relaxed about this. Here, it says 136, 312, 499 are natural numbers. But this trickster says integers. So, you can go with the common ones. Give 1 here, give 15 here, give 1 here, give 12 here, okay? But there are also negatives to these, you know, there are negatives to this too. If you say, " Okay, I'll do it for you, I'll solve it with mistakes," if I give 1, 15, 3, then 5, 3 divides 12, so 4, then if I give 5 here, it doesn't divide 12, it doesn't become 15, there are no other options, right? 15 already has 1, 3, 5, 15. 1 and 3 are only divisible by 12. 5 and 15 are not divisible. The sum from the first is 13, plus 15 is 28. The sum from the second is e. This is 7, plus 5 is 12. What will the answer be? Since it says k + m + n is the minimum, it's 12. The answer is not 12. The answer is -28. Why? Because it can be 1, 12, 15, but it can also be -1, -12, -1, -15. The product of -15 and -1 is 15, and the product of -1 and -1 is 12. Therefore, the total from here is -28. I don't need to write the other one, it's -12. Or write it if you want. Okay, I'll write it. If it will please you, take it. I wrote it. -5 -3. When you add them, k is -5, m is -3, n is -4. When you add them, it's -12. Actually, you see 20 as the answer, but it says integer. Okay, let me translate this, the smallest value here is -28. If they had asked the largest, I would have said +28. That's why we didn't cover number sets at the beginning for nothing; it's very important which set the number belongs to. Okay, okay, now that you understand the nuance here, let 's get to one of the important points. Look, these types of questions, or rather these equations, are very important. In equations given in the form ax + bx + c, you find the value that satisfies the equation. Then, this value is increased or decreased by the coefficients. These equations, Ax + by + C, will never be useful anywhere in your life, but I'll write them down anyway. They're called Diophantine equations, okay? We used them very often in university mathematics, in the abstract translation course, okay? Knowing the rule for these Diophantine equations is important. Now I'll apply the practice. The practice says find one value, then increase or decrease it by the coefficients of x and y. Don't try everything like a laborer. Now look, 2 A's + 3 B's = 35, right? Are there any problems here? Now look, what's the problem, sir? You wrote the exact same question, are you kidding us? Look, I'm going to find a value here that satisfies both A and B. But of course, I'll try to go from the smallest. Because if I find a value in the middle, like, "Let me give it 7," or "You're giving 7, but why not give 1?" Now, if you do this irregular value assignment, it becomes very difficult to count. You have to do this value assignment very regularly, okay guys? Otherwise, the question will blow up in our faces. Now, I'm going to assign values ​​to A and B. It doesn't matter if it's A or B. Let me give A the smallest value I can. It says A and B are positive integers, so the smallest value I can give A is 1, and I gave that. Look, when I move it to the other side, 3B becomes 33. This is 2. I moved it to the other side, 35 minus 2, so 3B becomes 33. Then B becomes 11. Is there a problem? Look, we found one value. Then, let me try 2 for A, that doesn't work, let me try 3, there's no need anymore. Now, you do it like this: you increase A because... You found the smallest value, you were increasing 'a', you put 8 on the other side, that doesn't work either, this comes out: 35 minus 8 is 27, 3B is 27, so what is B? 9. Look, why is B 9? Do you know, or how can we find this 9 the shortcut way? Remember how we increased 'a' by the coefficient of B? We also decrease B by the coefficient of A. What is the coefficient of A? 2. Decrease 11 by 2, 9. Now, I'm increasing 4 by the coefficient of B, and decreasing B by the coefficient of a. 7E, 7 also works, try it: 7K, 2, 14, 7 x 3, 21, 14, 21, add 35, isn't that right? Now, increase this by 3 more, 10, increase this by 4 more, decrease this by 2 more, 5, 10A, 5 also works, give 10, 20, give 5, 15, 20 + 15, 35, give 3 here, increase by 3 more, 13, decrease this by 2 more Reduce by 3, that gives 3 more, increase by 3, that gives 16, reduce by 2 more, that gives 1 more, it doesn't go any further. Because if you reduce 1 by 2 more, it becomes -1, not a positive integer. See? I was able to find all the others by finding just one. So what are you asking now, Dad? You tell us the difference between a and b, the values ​​of a, e, subtract 10 from 1, you get 11, -10, subtract 9 from 4, -5, 7, -7, 0, 10, -5, 5, 13, -3, 10, 16, -1, 15. In total, 1, 2, 3, 4, 5, 6. In total, friends, it takes 6 different values, okay? Look, I was able to solve the problem by finding just one and using the diopter equation property. Let's solve one more. It says, "A, b, and c are positive integers. If a, b, c, and 3a + 2b = 24, what is the minimum value of a + b + c?" Let's look now: 3a + 2b = 24. Let's start with the smallest value of A. By the way, we have to go from A. You don't have to, this time go from B, it doesn't matter. Give B 1, what happens? 22 becomes 2. I threw B, 3A, 22, A didn't come out as an integer, give 2. I threw 3A to this side, 4, 20, look, 2 times 2 is 4. I threw it to the other side, 20, 3A, 20 didn't come out as an integer, let's give 3, 3 x 2 is 6. I threw it to the other side, 24 minus 6 makes 18. 3A is 18, so A is 3A is 18, so A is 6. Damn, I couldn't divide 18 by 6, I was speechless for a moment, okay? Okay, we found one, the rest is simple. Now I'm going to increase B, increasing it by the coefficient of A, okay? I'm increasing B by the coefficient of A, I made B 6, and I'm decreasing A by the coefficient of B. I made this 4. Let's try it, 12, 12, 24, good. Now I'm increasing B again by the coefficient of A, by 3. I decrease 'a' by the coefficient of 'b', which also works for 2 to 9. Then I increase 'b' by the coefficient of 'a', making it 12. I decrease 'a' by the coefficient of 'b', making it 0. But that doesn't work, we can't give 'a' 0 because the question says a, b, and c are positive integers, so we ca n't. Therefore, this is out. Goodbye, Dad. So, A and BC are now... Look, when a, b, and c are 6, this becomes 3; when this is 4, this becomes 6; when this is 2, this becomes 9. Now we know that A is smaller than B. Therefore, the 6 to 3 possibility is eliminated because in the 6 to 3 possibility, it's not smaller than a, but larger, like 4 to 6 and 2 to 9. Now, C is the largest of these, meaning C is the largest of a, b, and c, but we need to choose the minimum sum. So, friends, here I will give C the smallest number greater than B. Since B is 6, the smallest number greater than 6 is 7. In the following possibility, since B is 9, it's 99... The largest and smallest numbers are 10. Now, let's do the addition in the first one. If I add these two or three, it makes 17. If I add these three, it makes 21. Which one is smaller? 17. So, our correct answer is 17. Okay, you've grasped it, right? This will come up a lot in questions. But from now on, instead of trying it out for so long, you can find all the values ​​by increasing or decreasing the coefficient of one by the same amount as the other. Okay? Now, friends, we've finished our comprehension questions related to this lesson, that is, the largest and smallest values, related to the first video of the second day. Now, we're putting a stop to the lesson with our purple question. What does the question say? Three different positive integers whose product is 990 will be written in the bottom three compartments of the circular compartments given below. The sum of the numbers written in two adjacent compartments is equal to the number written in the compartment above these two compartments. In other words, it says that the sum of these two is equal to this, the sum of these two is equal to this, and the sum of these two is equal to this. Accordingly, the largest value of the number written in the top circular compartment is equal to the smallest value. How much more than the value is this? Try this question, friends. It's not a very difficult question, but it has a very nice twist. There are quite a few different situations in the options. Now, let's say this number is A, this number is B, and this number is C. Their product will be 990. a x b ∩ C = 990. Now, I want to find the largest value of the top number, so I'll choose the furthest from each other so that their sum is maximum. Okay, let's choose one as 1. Then what do we choose for the other one? Since it says 3 different positive integers, let's choose 2. So we choose 495 so that their product is 990. Now, where you put 495 is very important. You would put it like this: 1 here, 2 here, 495 here, so this becomes 497, this becomes 3, and when you add them up, it becomes 500. I found quite a large one, quite a large one. Then you would say a x b x c equals 990. This time, the top number's product... By the way, let's make a note of this. Let's say we found 500, the largest value. Now, let's spread these numbers out so that their product is 990. This time, let's choose the closest numbers so that their sum is minimum. What comes to mind? 9, 10, 11, right? Multiplying them gives 90, 990, nice. So, I'm erasing this and now I'm placing them as 9, 10, 11. 9, 10, 11, what is this? 19, what is this? 21. 19 + 21 equals 40. Then, if you subtract 40 from 500, because you write in the circle above, how much greater is the largest value of the number than the smallest value? It seems like the answer is option A, but it's not. Because you're overlooking something here. Actually, I wouldn't scold my student for doing it this way. Because even if they don't write it down, after finding these numbers, we tend to arrange them here from smallest to largest. This is actually a problem-solving habit, but the question is nowhere to be found. Those three different positive integers we wrote below, from left to right or right to left, arithmetically from largest to smallest. It doesn't say they should be arranged from smallest to largest, so when you're finding the maximum value, if you write 495 on the far right, and 495 in the middle, and then write 1 here and 2 there, the sum of those two is 497, and the sum of those two is 496. See, in this sum, 400, 800, 180, plus 980, 900, 93, 993, 500, where is 993? That's why our largest value will be 993. The same thing applies to the smallest value, friends. If you move 9 here, 10 here, and 11 here, this is 19. This is... sorry, what is this part? 20. This is 19. When you add them up, you get 39. Look, we found a value smaller than 40, so the minimum value we can take is 39. Let's subtract them from each other, 993, 900, subtracting 39 gives 954. Our correct answer is option D. This is a habit in problem-solving. Okay, I wouldn't call it a bad habit, but we'll have to give up some of the standard things, be more alert, and never forget that ÖSYM (the Turkish examination board) always tries to trick us with these kinds of questions, okay? With a moral of the story, we've come to the end of this lesson with a good one. You know what happened? I finished the video, did the closing, reminded you about your homework, etc. Then, the idea of ​​starring questions came up. Right now, I've opened the video again. Starring is important for us, guys. It's both a summary of that day, or rather, that video, that lesson, and it helps you solidify those learning outcomes and make the learning process more immersive. That's why we need to star it. Where exactly? I'll put a star! I mean, this question deserves a star, God will turn it to stone! It's one of those questions that needs to be starred. Especially the part about taking the number in the middle and getting a bigger one – you need to pay very close attention to that. Guys, in diopter equations, you can star any one of them. I'll star this one because it's a bit harder than the others, but you can think about the others too. It doesn't deserve too many stars because it's a bit about the logic of the largest and smallest values... He understood. Pay attention to the common factors here. Solve it again. Be sure to pay more attention to the negative factors here. I want you to solve it again. His... In the light of E, and also those questions about the greatest and smallest values, if you just solve this one, you'll already have a complete grasp of the whole "I should give a larger value to the one with the larger coefficient, and a smaller value to the one with the smaller coefficient" thing. Solving these starred questions will be enough to ensure you have a general understanding of the topic, my dears. In the next lesson, friends, we will only cover one page from the VDK video textbook. Let me tell you right away, friends, we will only cover page 14, starting from example 15, and then we will finish. Here, we have 4, 6, or 7 concept examples. We also have a purple question where we put a dot on the greatest and smallest value. Now, let me remind you of the homework, and then I'll finish the lesson. What was the homework? What was the homework, friends? You are solving the " Greatest and Smallest Value Basic Definition Test 1" from the 70-Day TYT Mathematics Question Bank, that is, pages 12 and 13. There are 14 questions you need to solve here, friends. Solve them well. In the next lesson, I will give you the remaining homework, and we will have completed the greatest and smallest value section. Okay. I love you all very much. Take care of yourselves. See you in the second video, second lesson. Goodbye.