 All right, so let's look at problem 3.25. And again, you might wonder why I don't just show you how to solve this problem. And again, the thing to remember is if I show you how to solve a problem, it's because I don't think you're smart enough to figure it out on your own. So in general, I do think you're smart enough to figure it out on your own, so these videos are really unnecessary and they're produced because, well, I like to talk. But if you are really, really stuck on a problem, if you're really, really, really stuck on a problem, this video will give you some insight into how to solve the problem. Again, the thing to remember is that the opportunity to solve a problem is a once-in-a-lifetime opportunity. After you watch this video, any problem similar to this one isn't a problem. And you will never again have an opportunity to solve this problem. And the number of real problem-solving opportunities you have in any math course is extremely limited. If you're really, really, really, really, really, really lucky, your math course may have as many as 20, 25 real problem-solving opportunities. Most math courses have five. So let's take a look at this. Suppose I have two numbers and I know that they're divisible by something, and I know some more information. m divided by n is q with remainder r. And there's a bunch of questions, so we'll take a look at those questions, some of those questions, one at a time. So our first question is m minus qn divisible by d. So again, since we know the definitions, we can do mathematics. Again, I can't emphasize enough. You have to know the definitions. If you know the definitions, you can do mathematics. If you don't know the definitions, then whatever you're doing isn't mathematics. You are not doing mathematics if you don't know the definitions. So what do we know? Well, we do know m and n are divisible by d. That's what I suppose says. We know that m and n are divisible by d. So we know because this is what divisibility means, we know that m is d times something. We know that n is d times something. And there's some integers x and y involved in that. So because we know the definition of divisibility, because we know m and n are divisible by d, we know that m is d times something, n is d times something. Well, what about m minus qn? Well, I know what m is. Again, the equal says anytime I see this, I could replace it with this. And likewise, so m minus qn is, I'm going to replace m with dx. I'm going to replace n with dy, and I'm going to end up with this. Now, I can apply the distributive property of multiplication backwards. Here's the one tiny microscopic bit of insight that we need to know, which is that we can do a little bit of algebra. And we have our definition of divisibility. This thing, m minus qn, is d times something. And so that says m minus qn is also going to be divisible by d. Well, what about our next question? We want to know whether r is divisible by d. So, well, what do I know? I know that m divided by n is q with remainder r. And since I know the definitions, I can do mathematics. And so is our divisible by d? Well, the question is, can r be expressed as a product of d and something else? So, well, what do I know? Well, I know the definition of division. Since m divided by n is q with remainder r, the definition of division tells me something useful, which is that m is qn plus r. Well, I want to know something about r, so let's see if I can isolate r. And the definition of subtraction tells us that r is m minus qn. Well, wait a minute. I know from the previous question that m minus qn is divisible by d, and the equal says, if I see this thing, I can replace it with this thing. So I see this thing, I can replace it with r. We know that r is divisible by d. And so there's our conclusion. Well, again, now maybe I actually know what the remainder is. Well, what are the possible values of d? So I know that m and n are divisible by something. I know m divided by n gives us a quotient, and this time I know what the actual remainder is, so what about my possible values of d? Well, again, since we know the definitions, we can do mathematics. It also helps to remember what we've done before, so we can start with it. So by the previous question, we determined that the remainder has to be divisible by d, so we know that the remainder, 21, has to be divisible by d. So we know that 21 is going to be d times something, where d is some number. Well, what do we know? Well, we know that 21 is 1 times 21. So d could be 1. So here's a general rule in life. When you limit yourself to one answer, you limit yourself. And so, well, here I found what d could be. I can stop the problem here with one answer, but if I do that, I'm going to be limiting myself. So when you limit yourself to one answer, you limit yourself. Look at all possibilities. Open up to whatever the possibilities might be. So what else could it be? Well, I want to write 21 is a product of two numbers. So maybe a commutativity holds, 21 is 21 times 1. And so I know that d is a possible value. So d could be 21. Another possibility, 21 is 7 times 3. So d could be 7, or maybe d could be 3. And it never hurts to summarize the possible values of d, 1, 3, 7, or 21. And so there's my summary. What are the possible values of d? Could be any of these things. Well, let's flip the question. Suppose I know we found the possible divisors. Well, how do I know 11 is not a divisor of m and n? Well, here's an important observation. You already found all possible values of the divisors of m and n. So m and n are divisible by d. We already found all possible divisors. And so 11 isn't here. So we know that 11 can't be a divisor of m and n. It's worth emphasizing that if we stopped here and said, oh, d could be 1, we would have no idea whether 11 could or could not be a possible divisor. And again, when you limit yourself to one answer, you limit yourself. Now, the most efficient way of doing this is to actually find all the possible divisors of m and n, which is what we did. But let's say we didn't do that. So here's a different way of doing that. Here's another useful rule of life. If a decision takes you to a place you don't want to be, don't make that decision. Make the other decision. So again, since we know the definitions, we can do mathematics. And we have to decide whether or not 11 is a divisor of m and n. So, well, let's suppose it is. Suppose that 11 is a divisor of m and n. Then, because we know the definitions, we know that m is 11 times something and n is 11 times something. Again, since we know the definitions, m divided by n is q with the remainder of 21. We're told that. So we know that m is q n plus 21. And the equality says I can replace m with 11x. I could replace n with 11y wherever I see them. So that gives me m11x is q times n, q times 11y, 11qy, 21 stays 21. Rearranging things a little bit, I get 21 is 11 times something. And rearranging things a little bit more, 21 is 11 times x minus qy. And because I know the definition of divisibility, because I've written 21 as 11 times something, I know that 21 is divisible by 11. Wait, that's not true. 21 isn't divisible by 11. So that means we made a bad decision back here. This is not where we want to be. We don't want this outcome. 21 divisible by 11 is a bad result. So we must have had a bad decision back here. We made the decision 11 is a divisor. We have to make a different decision. 11 is not a divisor of m and n.