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we're on problem 132 if the integer n is greater than one is n equal to two so they tell us that the integer n is greater than one and they ask us is n equal to 2 statement 1 n has exactly two positive factors exactly exactly 2 pause factors well that's certainly true of the number 2 but it's also true of any prime number I mean and 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 there's definitely two statement 2 tells us the difference of any two distinct positive factors of n is odd this is interesting the difference of any two positive factors we're just dealing in kind of the positive world positive factors of n is odd so let's think about it we were you know just thinking about statement 1 we said okay what numbers have only two factors well prime numbers only have two factors so this is only this only tells us that n is prime right 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 they're odd right most prime numbers are odd let's look at other than 2 or the other prime numbers you go to 3 you go to 7 you go to 11 and 5 and 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 this not 2 is not divisible by 2 because it's prime and so they have to be odd numbers so for any other prime number the number is going to be 1 and itself so you know let's call that P so the difference between the two numbers and P is going to be odd number right every other prime number other than two is odd so the difference between the 2p minus one if I take an odd number and I subtract one from it I get an even number I get an even number that's true for any odd number so every other prime number is odd and this this happens but they're saying that when I subtract the difference I get a odd I get an odd number well the only number that that's true for is two because the factors of two or one and two and if I subtract one from 2 to minus one is equal to one and that's because 2 is the only even prime number so you this the 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 is if it's going to be fine let me think about the other two the other if statement two alone is enough difference of any two positive factors of n is odd 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 one and six right the Dafina one in six is five so it would satisfy this so n could be six if we just took statement two so we really need both statement one tells us that n is prime and statement two tells us that n has to be even right there has to be an even an even number so frankly there's there's only one prime even number and that's two so both statements together are necessary to answer this question 133 133 every num every member of a certain Club volunteers to contribute equally to the purchase price of a 60 dollar gift certificate how many members does a club have so we want to know is called mm4 members how many members of the club have question 1 each members contribution is to be $4 so the members so and let's see it says they contribute equally to a purchase price okay so the number of members times ever 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 and 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 - if five club members fail to contribute the share of each contributing member will increase by two dollars okay so that means that if I were to take the amount that five members were to contribute so let's say that the contribution amount is C right so if we take if we take the amount that five members would contribute so five times C and divide it by the remaining members so divided by M minus five that that would be that each contributing member will increase by two right so this is the amount that would have been contributed by those five people let me make sure that I'm not missing anything and that is equal to $60 that is going to be equal to two dollars right so this is the amount that those five can members would have contributed if they don't contribute it it's going to have to be divided by the other members so however members there are minus five and that is going to be when you divide the amount divided by who has to pay for it is $2.00 per leftover member and we also know that the members times the contribution for a member if everyone pays is going to be equal to $60 that they gave us in the problem statement right that M members of a are going to contribute equally and they're going to end up with $60 so actually have two linear equations in two unknowns so to a statement two alone is actually sufficient to solve this problem and if this doesn't look like a linear equation you can just multiply both sides by M minus five and you get five C is equal to 2m minus ten and now this looks a lot more like a linear equation and actually well actually this isn't a complete linear equation but let me let me solve it just to just a make the point clear for you so if I were to say that see this isn't a linear equation so I shouldn't have said that C is equal to 60 over m if C is equal to 60 over m so then this turns to 5 times 60 so 300 over M is equal to 2m minus 10 and 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 and then subtract 150 from both sides you get M squared minus 10m minus 150 is equal to 0 and then let me see if I can if I can just do this see just by factoring it minus 150 C 30 30 times 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 and 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 301 to 150 at 2m squared went to this had to be 5m that's my mistake 5m m squared minus 5n 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 so this was clearly not a this is actually a quadratic equation and we know that the members can't be negative right there's positive number of members so statement two alone is enough information to know that there are exactly fifteen members in this Club next problem that had me stumped there I could have my careless mistake there for a second next problem and I am okay problem 134 34 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 positive integers is a square root of n minus M an integer so m and n are positive integers okay so they tell us statement 1 they say n is greater than M plus 15 well I'd you know this n is greater than n plus 15 so if n is exactly 16 greater than M this says this tells us that n this is another way of saying that n minus m is greater than 15 right that tells us that what's under the denominator is greater than 15 well if what's under the Nama greater 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 n is equal to M times M plus 1 which the same thing is M squared plus M so if we substitute that into this equation we get the square root of M squared plus M plus M minus M right M Squared that that's M times M plus 1 it's N squared plus M and this minus M is right there so that cancels out we're just left with square root of M Squared which is going to be equal to M which is an integer so statement two alone is sufficient to say that this state this it would be an integer as long as n is equal to M times M plus 1 and I've run out of time see in the next video