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we're on problem where we 145 from 145 says is 1 over P 1 over P greater than R over R squared plus 2 that's their question statement number 1 I realize I go on these tangents before even looking at the statement I should look at the statements first P is equal to R so let's see if we can simplify this so this this if P is equal to R then we get 1 over R set of a P right is greater than R over R squared plus 2 let's multiply both sides of this equation by R so then we know that well we don't know that R is necessarily greater than 0 that's the problem we don't know because without knowing that R is greater than R so if R is greater than 0 well it's going to change the inequality one way or the other so if R let's just assume R is greater than 0 so if we multiply both sides equation by R then we don't have to switch the inequality because we're assuming R is greater than 0 and that's an assumption then you get 1 is greater than R squared over R squared plus 2 and then if R is greater than 0 than R squared plus 2 actually R squared plus 2 is always going to be greater than 0 because R square is going to be positive let's multiply both sides equation by R squared plus 2 and you get R squared plus 2 is greater than R squared you can subtract R squared from both sides and you get 2 is greater than 0 which is true but remember I have to make this assumption that R is greater than 0 if you assume that R is less than 0 R is less than 0 then all of this is going to break down because you're gonna get because when you multiply both sides by r you have to switch the inequality and you get 1 is less than R squared over R squared plus 2 and then you'll end up with well eventually you're going to get end up with r squared plus 2 is less than R squared or that 2 is less than 0 which is false right so this statement alone isn't enough we have to know whether or not R is greater than 0 statement two well there you go are is I really should look at boss statements first R is greater than zero so and just so you know statement two by itself isn't sufficient because if you know that are greater than zero you have no idea what P is so you can answer it yet but both statements combined are sufficient to answer this question next question 146 is an integer and integer is an integer statement one says N squared is an integer and squared an integer well that doesn't help us I mean you know what if n was equal to the square root of 2 and then N squared would be equal to 2 which is an integer but this is clearly not an integer but on the other hand and could be equal to 2 in which case N squared would be an integer so whether or not N squared as an integer both of these are cases where N squared or an integer but one ends up where n is an integer one is where and is it an integer so this by itself is not enough to tell me whether n is an integer statement 2 tells us statement 2 tells us square root of n is an integer ok so that essentially tells us that that n is equal to some integer squared right I mean you could take the square root of both sides you get the square root of n is equal to some integer so you take any integer you square it you're gonna get an integer so a statement two alone is sufficient to answer this question that was a strangely easy question they they were getting a little bit hairy and and confusing but but they've that was a little bit there was a nice little rest question I think 147 147 if N is a positive integer so n is a positive integer well n is greater than 0 it's an integer as well I didn't write that down is n to the third minus n n to the third minus n divisible by four fascinating fascinating in C so uh my initial reaction is to see if I can simplify this as the product of simpler expressions because I don't want to take I look already at statement number one let me look at statement number one statement number one says n is equal to 2k plus 1 where K is an integer I don't feel like kicking 2k plus 1 and cubing it you know that's you know that's not an easy thing to do it'll take some time so by adding my intuition is that maybe we should simplify this a little bit so if we factor out an N that's equal to n times N squared minus 1 and that's equal to n times what's n squared minus 1 that's n plus 1 times n minus 1 and now this is something that's much easier to substitute this into so let's do that let's substitute N equals 2k plus 1 into this so if n is equal to 2k plus 1 you get 2k plus 1 times 2k plus 1 plus 1 so that's 2k plus 2 that's this one and then you have 2 k plus 1 minus 1 so that's just 2 K and then what does that simplify to you get 2k plus 1 remember my I in the back of my mind I want it I want a 4 to show up because I want this thing to be divisible by 4 2k plus 1 times 4 K squared plus 4 K interesting so now I can factor out a 4 right so it's 2k plus 1 times K squared plus K times 4 and we know that each of these that this is an integer right because they told us that K is an integer where K is an integer we know that this is an integer because K is an integer and we know 4 is an integer so we've essentially kind of factored we factored N cubed minus N and four is one of its factors so it's definitely divisible by four I mean we can divide it by four right now if we divided by four we'd be left with this which is clearly an integer right because K is an integer so n the third minus n is divisible by four if we can assume that n is equal to 2 k plus 1 where K is some integer now what are they tell us in statement to statement to the statement 2 tells us N squared plus n is divisible by 6 well I don't see how that's how that's helpful at all I don't even n squared plus n is divisible by 6 I mean even how do you even you can't even relate to that to that I mean and N squared and I could try to do something fancy here no but this is just I mean this is so different than this N squared plus n is so different from n to the third there's no I mean I could factor out an N I could say n times n plus 1 is divisible by 6 well I guess that's interesting that tells us that this part is divisible by 6 n times n plus 1 is divisible by 6 that tells us well that tells us that they do they tell us that N is a positive integer yeah they do tell us that n is a positive integer so this could be 2 & 3 right if this is 2 & 3 then this would be 1 right this could be 2 & 3 in fact well this could be 2 & 3 but it could also be actually that's the only numbers it could be it could be 2 & 3 in which case this is 1 in which case this whole thing right and could be if you if you say this then n could be 2 and is equal to 2 n is equal to 2 n plus 1 is equal to 3 and plus 1 is equal to 3 and then n minus 1 would be equal 1 and so this whole expression let me see is that the only case I'm looking at the answer right now just going to make sure I wouldn't they say that this isn't sufficient but once again kind of like problem 142 actually because if you say that N is a positive integer if you say that N is a positive integer and that n times n plus 1 is the visit by 6:00 I can't think of any other numbers where you multiply one number times the next number and they're integers where you get a multiple of six oh well no no I take that back it could be three and four right there could be three and four it could be yeah there's actually a bunch of them so this it could also be you know this could be three this could be 4 this is a multiple of six and then this would be 2 so that's right this is not enough information so that's good because by this information n could be 2 n plus 1 could be 3 in which case this number would be actually 6 but that still doesn't help us because 6 isn't divisible by 4 but then I could come up with a situation where this number right here where n is 3 n plus 1 is 4 and n minus 1 is 2 where it is where this does become defensible by 4 so this doesn't give us enough information not useful and I'm out of time I'll see in 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