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we're on problem 151 151 if x and y are positive is the ratio of X to y greater than 3 so is X the ratio of X to Y is that greater than 3 and they're telling us that x and y are positive fair enough statement 1 X is X is 2 more than 3 times y so X is equal to 3 y plus 2 let's rewrite the original question because I think it's going to look a lot like this if you multiply Y times both side of this equation and we don't have to change the inequality because they told us that Y is positive they're both greater they're both positive so if you multiply both sides equation times y you get X is greater than 3y that was that's this is the same question as this well if we look at statement one number one clearly X is greater than three why because X is two more than three Y based on statement number one so statement number one is sufficient to answer this question X is definitely greater than 3 y X actually equals 3 y plus 2 this plus 2 tells you that X is that much greater 2 greater second statement the ratio of 2 X - 3 y the ratio of 2 X - 3 y is greater than 2 let's just do some algebraic manipulation 3y once again we know that's positive so when we multiply both sides of the equation by out we don't have to switch the sign so you get 2x is equal to is equal to 6y now we can divide both sides by 2 and you get oh sorry I didn't put it equal to be greater than greater than divide both sides by 2 and you get X is greater than 3 Y so this statement and this statement are the same thing so 2 alone is also sufficient so either statement alone is sufficient to answer this or this question 152 152 okay if arc PQ okay let me see if I can draw this so they've drawn a semicircle let me draw a circle that's a circle and I can make it a semicircle by drawing a line from there to there that's about as good as I can do and then they have it connecting to some point let me see if I can to do this looks like that and then it looks like that and then they actually have drawn an altitude coming down like that and I think I'm drawing done all my line drawing and then this let's see they label this P they label this Q they label this R and this is two and they say that this distance right here is a this distance right here and then they say that this distance right here is B fair enough so they say if arc PQ R so that's that arc right there is a semicircle what is the length of diameter PR so they essentially want to know what a plus B is right they want to know what a plus B is equal to what that's PR so that's their question so immediately well let's just look at the statements what did they give us statement one they tell us that a is equal to four so from knowing that a is equal to four if we can figure out what B is then we would know what a plus B is maybe we'd be done well let's see if we can do that statement two is B is equal to one so if we know what B is and then we could figure out a from it that we know what a plus B is so actually I think if either of these are sufficient the one the other one probably is because if statement one is sufficient if a if knowing a can we can figure out B then probably knowing B we could figure out a but let's see if that if that holds so one thing that I that kind of immediately should pop out of your head although I don't know if it will and this pops out of my head really from going to math competitions when I was a kid but if you have a triangle inscribed in a semi circle that triangle and one of its sides is a diameter right and that's the case with this one because we know this is a semi circle so we know that this is the ammeter then that triangle is a right triangle this is a right triangle and even the way they drew it it kind of looks like that but that's something you should know even if this triangle was drawn like that it's always going to be a right triangle and I'll maybe do another YouTube video proving that but that's just a good thing to know for the GMAT so let's see if we could use that information in the Pythagorean theorem to come up with a formula that relates a and B so if we look at just this triangle to me if we look at just this triangle right there we know what we know that a squared plus four is equal to let's call the side C squared is equal to C squared fair enough if we look at this smaller triangle right here we know that B squared plus four is equal to let's call this d squared that's the hypotenuse of the smaller triangle is equal to d squared now if we look at the larger triangle which we also known as a right triangle now C squared and d squared are the are the are the non hypotenuse sides and a plus B is the hypotenuse side so we know that C squared plus d squared is equal to this entire side squared which is a plus B squared well we know it's C squared and d squared are we can substitute that so C squared is a squared plus 4 and I'll switch colors just to avoid the monotony so C squared is a squared plus 4a squared plus 4 and D squared is B squared plus 4 right d squared is B squared plus 4 plus B squared plus 4 and all of that's equal to a plus B squared let's just multiply that out because I think we can do some canceling so we get a squared plus 2 a B plus B squared okay let's see so we can subtract a squared from both sides a squared a squared subtract B squared from both sides B squared and squared and this simples out simplifies out quite nicely you see four plus four is eight is equal to 2a B divide both sides by 2 and you get a B is equal to four so all of this crazy information that this is a inscribed in the semicircle and this is a this is B all of that reduces down to a B is equal to four so now we know if we know a if a is four then we know that B has to be equal to one and so and then we could figure out what a plus B is it's five so statement one alone is sufficient and similarly if you know what B is we know that a has to be four and so a plus B is equal to five so that was actually less hairy than I thought going into it next problem and this really is the homestretch 153 153 does the integer K have a factor P such that one is less than P is less than K does the integer K have a factor P such that okay so they're essentially saying is K Prime right this is another set way of saying is K or you could say is K naught prime K naught prime or K is a composite I think that's the word for a non prime number so that's all they're asking does K have some other factor other than one end itself so let's see statement number one K is greater than four factorial well four factorials four times three times two times one so that's 12 times two that's 24 so this ball is not decay is greater than 24 so I can give you both prime and non prime numbers greater than 24 29 is prime 30 is not prime so statement one at least by itself is pretty useless pretty useless you can K can be greater than 24 and be prime or not prime statement two this looks interesting 13 factorial which is a large number so it's hard to multiply it out plus two is less than or equal to K which is less than or equal to 13 factorial plus 13 so essentially what we need to prove somehow is that every number between this and this is not prime now let's think about how we can do it well let's take it let's let's take into the initial example of 13 factorial plus 2 so 13 factorial I'll just say you know that's 13 times 12 is 13 times 12 12 times 11 you keep going times 3 times 2 times 1 that's 13 factorial right so 13 factorial is definitely divisible by 2 so we could rewrite 13 factorial we could write that as we let's rewrite actually this 13 factorial divided by 2 times 2 plus 2 I just rewrote 13 factorial plus 2 and I want you to see that this is the same thing right I just divided by 2 and x 2 and you'll see in a second why I did that so and this is a tricky problem when I looked at this at first which which I actually did glance at before the video because it seemed a little bit it wasn't I had to think I don't know if this is something I would have gotten right if I was under time pressure on the GMAT but when I thought about a little bit more it made sense so if I divide by 2 and x 2 I could and this actually this is going to be an integer right because we know that 13 factorial is divisible by 2 2 is actually one of its factors right so as long as you understand where I got you know third you can divide anything by 2 and multiply it but it would not change it so this is the same thing as this right and then if we factor the 2 out if we factor 2 out this becomes 2 times 13 factorial over 2 plus 1 right I just factor the 2 out this is a this is a diffraction that's just a line I did just to not make it look too messy so what we show is that 13 factorial plus 2 is is definitely divisible by 2 right whatever that number is it's definitely divisible by two and how do I know that because this is just another way of writing 13 factorial divided by 2 I'm sorry this is another way let me write this down this is the same thing as 13 factorial plus 2 it is just another way of writing it I took 13 factorial divided by 2 and multiplying by 2 and then taking out the 2 and the reason why I know that this is that 2 is divisible into this is because I know that this is an integer we know that 13 factorial is divided by 2 because 2 is one of the things that goes into it so this is an integer you add 1 so this is an integer and so you're multiplying 2 times some integer so this has to be divisible by 2 so 13 factorial 2 plus 2 is divisible by 2 now let's just extend that generally we can we can 13 factorial plus X let's prove that as long as X is less than or equal to 13 that this is divisible by X so we can do the same trick we say this is equal to 13 factorial divided by x times X plus X that's equal to you can factor the X out x times 13 factorial over X plus 1 and how do we know that this number which is the same thing as this number how do we know that this is divisible by X well just a little thought experiment we know that as long as we know as long as X is greater than or equal to 1 and less than or equal to less than or equal to 13 and X is an integer as long as X is an integer that's greater than equal to 1 and less than or equal to 13 it is one of the factors of 13 factorial right 13 factorial is 13 times 12 times 11 is it's all of those numbers multiplied together so we know that X's as long as X is less than or equal to 13 we know that this right here this is an integer if this is an integer then the integer plus 1 is also an integer so 13 factorial plus X can be re-written as x times some some integer I'll call it I right but the bottom line is the reason why I wanted to do that it shows you that every number between 13 factorial plus 2 and 13 factorial plus 13 is actually divisible by whatever number you're adding to the factorial so since it's divisible by those numbers the numbers cannot be Prime so hopefully I've actually run way over time but hopefully I've shown you that that information alone statement two alone is sufficient to show that that K is not prime or that K does have out a factor P that's between 1 and K anyway I hope I didn't confuse you that was I think that was one of the hardest problems we've done so don't feel bad if you found that a little daunting see you in the next video