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we're on problem 99 and I like these that don't have a lot of words in them M does not equal end we have to prove that is it true that M does not equal n statement number one says that M plus n is less than zero well let's see what does this tell us this just tells us that M is less than negative and it still doesn't tell us anything I mean you know maybe negative n maybe negative n is well let's say if M is equal to n which is equal to 10 or is equal to negative 10 then this would hold true right because negative 10 plus negative 10 is minus 20 which is less than zero so this still doesn't prove anything or of course I could have two different numbers I could have M being -5 and n being -1 and when you add them together you get minus 6 both less than 0 so it still doesn't tell us anything whether M does not equal n statement 2 M times n M times n is less than 0 now this is interesting M times n is less than 0 so that tells us that one is positive and one is negative right the only way you can multiply two numbers and get something less than 0 is 1 positive 1 positive and 1 negative right and it doesn't say equals 0 so that you both can equal 0 and and so when you multiply two numbers and you get a negative number they have to be different numbers because one has to be positive and one has to be negative try it out with as many things as you want but you know from your basic negative multiplication rules that the only way to get a negative as the product of two numbers is one positive one negative so this is all we need to know to know that m and n are different because once positive and ones negative I said that about five times problem 100 and you didn't need statement one problem 100 when a player in a certain game tossed a coin a number of times for more heads than tails resulted so heads is equal to tails plus four heads or tails resulted each time the player toss the coin fair enough how many times did heads result so the essentially the the problem description just tells us that heads is equal to tails plus 4 and that heads plus tails is just all of the all the tosses so statement number 1 says the player toss the coin 24 times so that's what I just said heads plus tails is equal to 24 and so this is enough information by itself to know how many times did head result because we could rewrite this top equation as heads - tails is equal to 4 and if you wanted to solve it you'd get two heads it's equal to 28 heads is equal to 14 all of that would have been a waste of time as soon as you saw that you had two linear equations and two unknowns you could said oh I have enough information to solve for any of the variables right this is a four all right so statement 1 by itself is enough statement 2 the player received 3 points each time heads resulted and 1 point each time tails resulted for a total of 52 points so got 3 points for every head so 3 that's how many points you got for all the heads plus 1 point for each tails so that's how many point you got for all the tails and that equals 52 so once again this is providing us with another linear equation so now we have two linear equations with two unknowns I'm not even looking at at what they gave us in statement 1 so just using these two we have enough information to solve the problem you could do it the exact same way this is what you learn in algebra 1 so each statement independently is enough or as is sufficient to solve the problem problem 101 we've already done 100 problems 101 once you get going these get kind of fun 101 if s is the infinite sequence s 1 ok so it's like s 1 is equal to 9 s 2 is equal to 99 s 3 is equal to 3 nines I get it and the the KS is equal to write 10 to the K minus 1 right this is a third one 10 to the third is a thousand minus 1 10 squared is 100 minus 1 right so it all works all right that's what they told us is every term in s divisible by the prime number P divisible by P well it depends what P is right and frankly if if just the first term is divisible by P all of them are going to be divisible by P why is that because all of the terms are divisible by the first term or 9 right so if P goes into 9 it's gonna go into all of these terms because 9 goes into all of them right so we just have to see say does P go into 9 it's a much simpler way of thinking about the problem statement number one says P is greater than 2 so does every prime number does every prime number greater than 2 go into 9 well no 5/7 is don't go into 9 3 goes into 9 so we still don't know whether P goes into 9 statement 2 at least one term in the sequence s is divisible by P at least one term at least one term divisible by P if we knew that it was this term if we knew that the first term was divisible by P we would be all set right but we don't know whether it's it's just the first term for example P could be 11 what if P is 11 if P is 11 then it goes into this term it goes into the 999 but it won't go into the first term it won't go into 9 right so you know 11 is an example where it holds for case 2 but it doesn't hold for the whole question it's not divisible into every term or we could say that if P was equal to 3 well of course that will go into every one so statement 2 by itself actually doesn't help eseni and actually both of these numbers satisfy both statements and so if you pick 11 it doesn't work if you pick 3 it does work so both statements combined still do not give us enough information problem 102 I got to do some drawing okay so I have a quadrilateral here let's see it's ru and then go up there go flat to there and then come down like that and then they draw an altitude here already okay so this is an R this is aw this is au a T and an S they're saying the height of this is 60 meters this length is 45 meters and they're saying this right here this length is 15 meters fair enough okay quadrilateral rst you shown above is a site plan for a parking lot in which side are you is parallel to side st okay so this and this are parallel what is the area of the parking lot so immediately I can figure out the area of this triangle right base times height times 1/2 so I can figure out this area immediately I can immediately figure out I can immediately figure out this square region right here because I know it's 60 by 45 and so the question is can I figure out this triangular reason right here well I don't know what this distance is if I knew what this distance is and I could say base times 60 times 1/2 and figure this out so this is this right here is what matters thus that's the crux of the issue right there so statement number one tells us that our U is 80 meters our U is equal to 80 I think this gets us there right because think about it if our R if this whole thing is 80 meters if that whole thing is 80 I can figure out that how can I figure out that because I'm 15 here I have this length is 45 so this length going to be 80 - 45 - 15 oh it is that 60 so 80 minus 60 so this will be 20 and so just what statement 1 I can figure out this length right here and I'm done because this triangle is this base times this height times 1/2 this rectangle is this base times this height and then this triangle is 20 times 60 times 1/2 so I have enough information to figure out the area we just with statement 1 now statement - tu tu is equal to 20 square roots of 10 meters tu its 20 square roots of 10 well this is enough information to think about it this is a right triangle right here this height is 60 and remember if we can figure out let me switch colors if we figure out the base we're going to be set so we could do Pythagorean theorem so we could say the Bay squared plus 60 squared is equal to this number squared and that number squared is what 400 times 10 right 400 times 10 so it's 4,000 so you get V squared is equal to 4,000 - 3600 is equals 400 so V is equal to 20 again so this actually also gives you the information that this this base right here is equal to 20 and then like I showed in the previous statement that's all you need to figure out the area of the entire parking lot so statement 2 alone is also sufficient so either of these statements alone are sufficient to tell us the area of the parking lot see in the next video