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we are on problem number 22 and their question is straightforward enough what is the value of the integer X and they tell us it's an integer X is equal to what so the first statement X is prime well there are a lot of prime integers there's actually an infinite number of them so that alone doesn't tell us what X is 2 X is let's see 31 is less than or equal to X which is less than or equal to 37 well immediately this is useless information because 31 and 37 are both prime numbers as far as I can tell right they're both prime numbers so this means that X could be 31 it could be 37 or let's see are there any prime numbers in between 32 33 34 35 well the no prime numbers in between but could be the 31 or 37 so it doesn't help us much even in used in combination I mean this alone is definitely useless because this is a range of six numbers and even if we know X is prime if we use this and this all we know is that X is either 31 or 37 so the answer is e that they're together they're not sufficient e now if this was if this was a if this was a little bit different if this was X is greater than 31 and less than or equal to 37 then we would know Oh X has to be 37 or if it was if this wasn't here and this was there then we would know that X is 31 but anyway they didn't do that they let us gave us the option so the answer is e problem 23 23 if PQ and r are three distinct points do line do line segments PQ and PR have the same length so P Q and R I want to know if these two lines are the same length this is the line PQ and then this is the line PR so are those the same length let's see if we can figure this out statement one tells us P is the midpoint of line segment QR okay P midpoint so I've already drawn it wrong midpoint of Q R so let me redraw it so it actually that they're all they're actually all three points actually end up being on the same line because P is the midpoint of Q R so if this is the line this is the line this is point Q this is point R so if P is the midpoint P is right there and that tells us that this distance is the same as this distance which also tells us that QP line segment QP is equal to Q is equal to P R and that's what they'll I think what they wanted to sort yet right they want to say does PQ and PR have the same length this is P Q so P Q and P are definitely have the same length so statement one is definitely true because P is the midpoint let's see what statement to and I mean statement one implies that they have to be on the same line otherwise otherwise if P couldn't have been the midpoint let's see if statement two is interesting statement two Q and R lie on the same circle with Center P interesting so this is saying so a circle a circle with Center P Center P and Q and R both lie on this circle so Q could be here I'm just picking two random points are could be here so we want to know whether Q whether P Q is equal to P R right that was the question well the definition of a circle R is all points that are equal distant from P so R has to be the same distance from P is going to be one radius of the circle away from P as Q is because they're both on the circle and they're all equal distance from P so P R is going to be equal to PQ they're going to be equal to each other so once again two alone is sufficient so the answer is D each statement alone is sufficient problem 24 the problems are getting a little bit more interesting problem 24 is the number X between 0.2 and 0.7 so that's how we can so if I just write at Point 2 is less than X which is less than 0.7 that's what they want to know statement 1 tells us that 5 60 X 5 60 X is less than 280 and you know you don't have to solve this you just know that if this is a fairly simple equation you just can easily solve for X and if you can solve for X you can answer this question so you actually don't have to go through the trouble if you want to you can then just divide both sides by 5 60 but that takes time valuable time when you're taking the GMAT and that is equal to 1/2 right 28 over 56 right is 1/2 and then you would say sure X is between the two numbers but you didn't have to do this that's a waste of time problem 2 all you have to do is say oh I could solve for X so I can test that whether that's true or not part 2 statement two says 700 X is equal to 700 X is equal to 280 same thing you just look at you like oh I could solve for X if I can solve for X I can tell you whether X is between those two numbers and that's the end of it and you would say oh each of these independently are sufficient to answer this question but if you wanted to prove it to yourself and see if this question is true or false you can just say well X is equal to what 280 over 700 that is equal to what is that 2.8 over 7 that is equal to 0.4 I just divide the top and the bottom by 7 point 4 so once again it is between it you got you answer the question but you remember the question doesn't have to be affirmative just have to figure out if the data is enough to answer the question and we didn't have to go through this and this you could have just looked at and said oh I can solve for X in either way and then just test it next problem 25 I'll do it in moe 25 if I and J are integers is I plus J even I plus J even so the statement number 1 is that I is less than 10 that tells me nothing I that add to me that that seems kind of useless statement number two is I is equal to J I is equal to J well that's interesting so what's I plus J then going to be equal to well that's the same thing as since I is equal to J that's the same thing as J plus J which equals to J or you could say you know if you substitute J for I or if you substitute I for J that's the same thing as I plus I which is equal to 2i either case you immediately see that either of these numbers are multiple of two and therefore have to be even because I and J are integers so statement two alone is sufficient and statement one is useless problem 23 let me go to a brighter color no 26 from 26 we're on 26 n plus K is equal to M what is the value of K so we could solve for it we if we could answer this because K is equal to M minus n that will solve our answer let's see if one if the statements give us any help n is equal to 10 N is equal to 10 well that'll just tell us that K is equal to M minus 10 it still doesn't tell us what K is because we don't know what M is statement number 2 M plus 10 is equal to n this is interesting M plus 10 is equal to N so this tells us that what does this tell us so let's see if we can if we can figure out from this statement what M minus n is then we're in business so let's see let's subtract n from both sides you get M minus n plus 10 is equal to 0 subtract 10 from both sides you get M minus n is equal to minus 10 well we know that K is equal to M minus N and minus n is equal to minus 10 so this is equal to K so we solved for K using just statement two alone just statement two alone is sufficient and statement one is I mean it kind of gets this half way but you don't really don't need it statement two is all you need so that is B see if we have time for one more yes sure why not they have drawn a triangle for us I'll draw it quick and dirty see and this is angle X this is angle Z and this is angle Y and they want to know is the is this triangle equilateral which means and are all the sides equal to each other so statement one tells us that X is equal to Y so we break out a little bit of our eighth or ninth grade geometry here so if X is equal to Y if these angles are equal to each other what that tells us is that these sides are equal to each other the corresponding sides are the same I can actually redraw this triangle like this maybe it'll jog those neurons from ninth-grade if this is X this is y and this is Z and you could test it out if you don't believe me and we've gone over this into the geometry module if x and y are equal to each other then these sides have to be equal to each other and you could just try to imagine drawing a triangle that has two base angles equal to each other and the sides are different because it would be impossible the sides would have to be the same this triangle would have to be at least isosceles and and maybe equilateral isosceles is two sides being the same but that by itself doesn't tell us that this is an equilateral triangle in order for this to be equilateral z has to be the same as X&Y they all have to be and if they're all going to be the same they all up to 180 they'll have to be 60 degrees so you know you could have a situation where x and y are both I don't know 20 degrees if this is 20 this is 20 then they add up to 40 and then Z would have to be 140 so this doesn't and in which case we would not have an equilateral triangle so what alone is not sufficient let's see what statement 2 tells us statement 2 tells us Z is equal to 60 Z is equal to 60 degrees so that by itself so first of all we know if we have if we know both of these pieces of information we are definitely dealing with an equilateral triangle because think about it if Z is 60 is Z is 60 so if we use this and then we use this information know that these two are equal to each other what do we know if Z is 60 then these two added together have two so X plus y plus Z are going to be equal to 180 if Z is 60 then X plus y have to be equal to subtract Z from both sides or subtract 60 from both sides has to be equal to 120 and since these are equal to each other x and y both would have to be equal to 60 degrees right so if you use both of them for this information all of the angles are going to be 60 degrees and it's equilateral so we know that definitely come in combination we can figure it out but what about just if Z is equal to 60 well I can easily draw a triangle that disproves that so if I have a 60 degree angle here in fact it might be a 30-60-90 triangle like you we learned in school so you could be 60 degrees 30 degrees and 90 degrees and this could be Z and this definitely is not an equilateral triangle and so if just by Z being 60 degrees in no way are you saying that this is definitely an equilateral triangle you have to have the other condition that X is equal to Y and so the answer is let's see see both statements together are sufficient but neither statement alone is sufficient so the answer is see see in the next video