We're on problem 132. If the integer n is greater than 1, is n equal to 2? So they tell us that the integer n is greater than 1, and they ask us, is n equal to 2? Statement 1, n has exactly two positive factors. Well that's certainly true of the number 2. But it's also true of any prime number. I mean n could be 7. 7 only has two positive factors, 1 and 7. So it's true of any prime number. So this, by itself, isn't enough information to say that it's definitely 2. Statement 2 tells us the difference of any two distinct positive factors of n is odd. This is interesting. We're just dealing in kind of the positive world. So let's think about it. Just thinking about statement 1, we said, OK, what numbers have only two factors? Well prime numbers only have two factors. This only tells us that n is prime. 2 is a prime number, but it's not the only prime number. But for most other prime numbers, what do we know about them? Well they're odd, right? Most prime numbers are odd. Other than 2, what are the other prime numbers? You go to 3, you go to 7, you go to 11, 5. Let me ask you, why are they odd? Because if they were even, they would be divisible by 2 and they wouldn't be prime. So by definition really, every prime number that's not 2 is not divisible by 2 because it's prime. So they have to be odd numbers. So for any other prime number, the number is going to be 1 and itself. So let's call that p. So the difference between the two numbers, and p is going to be an odd number. Every other prime number other than 2 is odd. So the difference between the two, p minus 1. If I take an odd number and I subtract one from it, I get an even number. That's true for any odd number. So every other prime number is odd. And this happens. But they're saying that when I subtract the difference I get an odd, I get an odd number. Well the only number that's true for is 2. Because factors of 2 are 1 and 2. If I subtract 1 from 2, 2 minus 1 is equal to 1. That's because 2 is the only even prime number. So this first statement says we have to be dealing with a prime number. The second statement says, well, the number really has to be an even number if it's going to be prime. Let me think about the other two-- if statement 2 alone is enough. The difference of any two positive factors of n is odd, any two positive factors of n. Now this alone doesn't help us because n could be-- I don't know-- it could be 1 and 6. The difference between 1 and 6 is 5, so it would satisfy this. So n could be 6 if we just took statement 2. So we really need both. Statement 1 tells us that n is prime. Statement 2 tells us that n has to be even, that it has to be an even number. So frankly, there's only one prime even number, and that's 2. So both statements together are necessary to answer this question. 133, every member of a certain club volunteers to contribute equally to the purchase price of a $60 gift certificate. How many members of does a club have? So we want to know-- let's call m for members-- how many members does a club have? Question 1, each member's contribution is to be$4. Let's see, it says they contribute equally to a purchase price. OK, so the number of members times-- and they say every member of a certain club. So now they don't say some members. So m is the number of members, and they contribute equally $4. That's going to be equal to$60. So then you immediately know that you can solve for the number of members. There are 15 members of this club. Maybe I'm missing something. Statement 2, if 5 club members fail to contribute, the share of each contributing member will increase by $2. So that means that if I were to take the amount that 5 members were to contribute-- so let's say that the contribution amount is c. So if we take the amount that 5 members would contribute, so 5 times c, and divide it by the remaining members, so divide by m minus 5, that would be that each contributing member will increase by$2. So this is the amount that would have been contributed by those 5 people. Let me make sure that I'm not missing anything. That is going to be equal to $2. So this is the amount that those 5 members would have contributed. If they don't contribute it, it's going to have to be divided by the other members. So however many members there are minus 5. That is going to be-- when you divide the amount divided by who has to pay for it-- is$2 per leftover member. We also know that the members times the contribution for member if everyone pays, is going to be equal to $60. That they give us in the problem statement, that m members are going to contribute equally, and they're going to end up with$60. So actually, we have two linear equations and two unknowns. So statement 2 alone is actually sufficient to solve this problem. If this doesn't look like a linear question, you can just multiply both sides by m minus 5 and you get 5c is equal to 2m minus 10. Now this looks a lot more like a linear equation. Well, actually, this isn't a complete linear equation. But let me solve it just to make the point clear for you. So if I were to say that c-- this isn't a linear equation, so I shouldn't have said that-- c is equal to 60/m. If c is equal to 60/m, so then this turns to 5 times 60. So 300/m is equal to 2m minus 10. Then you are left with what? Multiply both sides by m, you get 300 is equal to 2m squared minus 10m. Divide both sides by 2, you get 150 is equal to m squared minus 10m. Then subtract 150 from both sides. You get m squared minus 10m minus 150 is equal to 0. Then let me see if I can just do this just by factoring it. Minus 150, 30 times 5, no, that's not good. 15 times 10. So if I do m minus 15 times-- no, that doesn't work. 15, 25, and 6. I know one of the answers already. I know it's 15. m minus 15 times m plus 10. Oh, actually, I just realized what my mistake was. I went from this step to this step. So I divided both sides by 2. So 300 went into 150. 2m squared-- this had to be 5m. That's my mistake. 5m. M squared minus 5m minus 150. So that's m minus 15 times m plus 10 is equal to 0. So that tells us that m is equal to 15 or minus 10. This is actually a quadratic equation. We know that the members can't be negative. There's a positive number of members. So statement 2 alone is enough information to know that there are exactly 15 members in this club. Next problem. It had me stumped there because of my careless mistake there for a second. Next problem. OK, problem 134. So that last one, I don't know if I just said it. Each statement alone is sufficient. So problem 134. If m and n are positive integers, is the square root of n minus m an integer? So m and n are positive integers. OK, so they tell us. Statement 1, they say n is greater than m plus 15. Well you know, n is greater than m plus 15. So if n is exactly 16 greater than m, this is another way of saying that n minus m is greater than 15. That tells that what's under the denominator is greater than 15. Well, if what's under the denominator is 16, then it is an integer. But if it's 17, which also meets this requirement, it's not an integer. So this isn't enough information by itself. Statement 2 says n is equal to m times m plus 1, which is the same thing as m squared plus m. So if we substitute that into this equation, we get the square root of m squared plus m minus m. That's m times m plus 1. It's m squared plus m. This minus m is right there. So that cancels out and we're just left with square root of m squared, which is going to be equal to m, which is an integer. So statement 2 alone is sufficient to say that this would be an integer as long as n is equal to m times m plus 1. I've run out of time. See you in the next video.