Integer Pairs That Balance Perfectly

Hello everyone. In this video we're going to be solving a Diophantine equation, which means we're going to be solving an equation for integer solutions, although you may be more interested in positive integer solutions. So let's go ahead and take a look at this problem from different angles. First method. I'm going to go ahead and cross multiply because why not, right? If you do that you're going to get something like xy equals x plus y times 72, which is something you can distribute, and that will give us xy equals 72x plus 72y. So can we find a number, or two numbers actually, integers x and y, such that this equation is satisfied? That's what we're looking for. But how do you go about solving it? Well there's a really cool method called Simon's favorite factoring trick, which is also abbreviated as SFFT, or I just call it Simon. We can use it. So here's how it goes. I'm going to subtract everything on the right hand side, and of course that sets everything equal to zero. Now we're going to focus on the first two terms from left to right. So I'm going to go ahead and try to factor this, but of course some terms are missing. So I need to find out what do I need to add or subtract. So if you take out x you're going to get y minus 72. Beautiful. Now I need to continue with the same thing, but notice that the coefficient of y is minus 72. So after the minus sign I need to put a 72 there, and of course I would like to continue using the same common factor y minus 72. So what if you wrote it there? Pretend you have the minus 72 there, but that would involve the product of these two things, which is 72 times 72. You can write it as 72 squared for now. Later on we may simplify it. Okay 5,000 something probably, right? Now let's go ahead and notice that we have a common factor. So we can factor it out y minus 72 times x minus 72 equals 72 squared. So I'm going to show you how to find solutions. We may not go through all the solutions because once you know the technique, I mean the rest is kind of straightforward. So here's the idea. If x and y are integers and you subtract 72 from them, they're still going to be integers, right? We didn't add square root of two or we didn't divide by any number. So these are still integers and their product is another integer. So can we find two integers whose product is an integer? Absolutely. That's called factoring. How do you factor 72 squared? There are so many ways to do it, but let me show you a couple of them. For example, one way to do it is 72 and 72 because 72 times 72 is 72 squared, right? You agree? Okay, great. So now this gives us the following. x minus 72 equals 72. That means x is equal to 144 and y is equal to 144. Is this really true? I mean if you take 144 and multiply it by itself and then in the denominator you add them, is this actually going to equal 72? Let's find out. Look, this is what we can do. The denominator can be written as 2 times 144 because you're adding the same thing twice. 144 cancels out, one of them, and this becomes 72. There you go. So our answer checks and we don't have to do this every time, but I just wanted to show you once, you know, so you can be more confident. Now this is one of the solution pairs. There are more, obviously, because 72 squared can be factored into other numbers, such as what? Well, to understand how this goes, we probably want to look at the prime factorization of 72 squared. Well, if you can do 72, then you can do 72 squared. It's just going to be 8 times 9 or 2 to the 3rd times 3 to the 2nd, and now we have to square that. The In other words, 72 squared is the same thing as 64 times 81, but you didn't know that, did you? Come on, be honest. Well, I didn't. So here's what we're going to look at. How do you find the factors of a number or the number of factors of a number? Well, look at the exponents. 2 to the 6 from 2 to the power 0 to 2 through 2 to the power 0 through 2 to the 6. It can appear in 7 different ways. We can have a 2 to the power 0, we can have a 2 to the 1st in the factors, and this gives us 3 to the power 0, 3 to the 1st, 3 to the 2nd, 3 to the 3rd, and 3 to the 4th. There's 5 options here, and we have 7 options here. It's all about options, and now when you put these two things together, for every choice from a power of 2, you get to choose one of these threes. So together, multiplication principle tells us combinatorics that 7 times 5 is the option. So this number has 35 factors, which means we're going to have to look at 35 pairs, but of course they're going to repeat because of symmetry, so you don't have to do all of them. For example, here x and y are equal, and let's say another factor that you want to look at. How do you find which factors to look at? Let me show you. This time I want to write the x first, that's alphabetical order. Now notice that this is equal to 72 squared, which I wrote as 2 to the 6th times 3 to the 4th. So here's what you can do. You can pick some twos and threes from here, anything you like. Let's say I just want to use 2 to the 6th, which is 64, and the other one is going to be 3 to the 4th, which is 81, and then now I got a pair. Make sense? And if you add these, you're going to get x equals 136 and y equals 153. Is this really going to work? Absolutely, because it comes from here. But there are so many other ones. Why? Because, again, because of the way we can factor, there's so many different ways you're going to find a lot of things. But here's the thing. Once you pick one factor, they automatically have to pick the other one for the other number. Make sense? I hope this does make sense. And for the negative numbers, let me just show you one, because that's kind of like the easiest. And I'm not sure if it's going to be negative though. But anyways, you get the idea. So you want to get 72 squared, but you want to use negative factors. And that can be done, because negative numbers times a negative number is positive. You know that, right? So this can be negative 72, and this can be negative 72, and you still get 72 squared. But that means x and y are both zero. Uh-oh, Houston, we have a problem. You can't divide zero by zero to get 72. So this solution, obviously, is not going to fly. I mean, it's not going to work, as you know. But why did it come up? Because we changed the equation. We changed the domain, the solution set, when we cross multiply. Because we're assuming x plus y is not zero, which is not true if x and y are both zero. Make sense? Okay, let's go and look at alternative ways. I haven't really thought about it. And sometimes, you know what? I don't usually have a script. And some people complain about, like, you shouldn't have a script. You shouldn't make these mistakes, whatever. You know why I don't have a script? First reason, I'm lazy. Second reason is, it's more natural that way. So you kind of get to see the problem-solving process, my thinking process. And sometimes I get stuck, and you get to see that too. And I make mistakes, and I mess up things, and I usually don't cut them unless they're really, really, like, messy or critical, whatever. And so you get to see that as well, because not everybody is perfect, right? Well, maybe nobody's perfect. Okay, anyways. So a second way to approach this problem could be something like this. You could try to solve for y. How? Well, first of all, you don't really have a choice. You kind of have to cross multiply here. And then since you're trying to solve for y, why don't we bring these two terms together? Factor out y, you're going to get x minus 72, and then 72x will be on the other side. Now we can go ahead and isolate y here. But of course, at this point, you kind of need to do a little bit of focus, focus, okay? But I'll show you the trick, the mathematical trick, okay? Here's how it goes. We want to get this, right? I mean, what do I mean by that? I mean, we kind of need to make the numerator divisible by the denominator. Does that make sense? Okay. So forget about the 72 for now, because that'll complicate things. Why don't we just focus on this? Can we do that? Absolutely. Look, I can just subtract 72 and then add 72. Boom, we're even, right? So this works, and this is good, because now I can split it up, and this will give me one, as you know. You should be quick at these things, but just keep practicing, guys. You're going to get better and better. It really sucked. I got to tell you something. When I was about to graduate from university, I was kind of dealing with problems like this, because I always solve problem solving. And there's another friend of mine who's currently in Texas, by the way. Anyways, if he watches this video, I just want to say hi to him, hopefully. But anyways, we were just solving some problems, right? And we sucked at solving problems. We were really bad at it, because we were so new. But what happened was, I mean, we made a lot of mistakes. We didn't know how to proceed, but you know, keep practicing, and you're going to get better. That's my point. I don't know if I explained real well, but you get the idea. So anyways, so you see what I'm talking about? That's kind of like broken down, but this is just that part. So you're gonna have to multiply this by 72, but that's easy. Look, it's really clean. Now we're gonna go ahead and distribute, and it's gonna give you the same thing, but in a just different way. Now here's what you need to think about. In order for this to be an integer, like y needs to be an integer, so x minus 72 must divide 72 squared. So we get to the same point. If you subtract this and cross multiply, we get the exact same thing. So is there a third way to do this? That's a good question, right? Well, at least I can come up with a couple solutions. So let me go ahead and show you what I'm thinking, and if you know of a better way to approach this, I mean, besides anything I did, please let us know in the comment section, and also let us know what you think about the video. Positive or negative, doesn't matter. Don't be too negative, okay? Anyways, so what can I do? Well, here's what I can do. I can flip this, because when I do, it's separable, right? You see what I'm talking about? Now this can be written as x over xy, which is 1 over y, and then 1 over x. Awesome. So we're adding two unit fractions, and we're getting a unit fraction. You know, wait a minute, are these called unit fractions or Egyptian fractions? Something like that. Okay, correct me if I'm wrong, but here's the idea. We want to add two fractions whose numerators are 1, like 1 over something plus 1 over something, and we want to get 1 over 72. Guess what? 1 over 72 can be written as 2 over 144. So if you add 1 over 144 and 1 over 144, you get 2 over 144, which is 1 over 72. So x and y can be these guys, and we already found them, remember? That was our first solution, right? But can we find more solutions? Sure, maybe you can try expanding like differently, like instead of multiplying by 2, can you multiply by 3, maybe? This will give you 3 over 216, and maybe you can break it down into 1 over 216, but the other one has to be 2 over 216. Oh, we're lucky, because this is divisible by 2, so I can simplify, and ta-da! I get another solution, which we didn't get before, remember? But continuing in this manner, that's going to take forever. I don't recommend. That's why it's the third method. Maybe it should be the first, but anyways, this brings us to the end of this video. Thanks for watching. I hope you enjoyed it. Please let me know. Don't forget to comment, like, and subscribe. I'll see you next time with another video. Until then, be safe, take care, don't forget to check out A plus BI, and bye-bye!