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Current time:0:00Total duration:10:13

we're on problem 148 148 what is the tens digit of positive integer X we one of the tens digit tens digit statement one tells us X divided by 100 has a remainder of 30 well then we know the tens digit is 3 right because let's say 30 divided by a hundred has a remainder of 30 and it's tens digit is three 130/100 has a remainder of 30 and its remainder is 3 and I can make an argument that any number that when you divided by a hundred that has a remainder of 30 is tens digit is going to be three in fact that number is going to be three zero and then it's going to have a bunch of other digits here there there and if you just divide this by 100 you're going to be left over with 30 so statement one is sufficient to figure out what the tens digit of the number is statement 2 X divided by so we could write X divided by 110 has has remainder equal to 30 well there's a little different case so definitely when 30 is divided by 110 the remainder is 30 because it goes into it zero times so the remainder is 30 and then the tens digit is 3 but let's see I'm gonna try to find other numbers when they divided by 110 its remainder is 30 but it's tens digit is something else let's see 140 if I divide that by 110 I'm left with a remainder of 30 of of 30 but it's tens digit is what its tenth digit is for right I shouldn't draw an arrow here if I did let's say I go up one more 250 250 you divided by 110 its remainders 30 but it's tensage is 5 so this statement gives me no information saying that the remainder story still gives me no information the Tencent can be any of these numbers or a bunch more if I just kept going so statement one is sufficient to answer this question question 149 149 if X Y & Z are positive integers is X minus y odd in order for this to happen one of these have to be odd and one have to be even and you can just think about that if you take the difference or really the sum of two numbers the only way that that difference or sum is going to be odd is if one is odd and one is even so let's think about that let's look at the statements it's gonna see if it gives any information they say that X is equal to Z squared well that still doesn't give me well it gives me no information about Y and it gives me actually very little information about X just yet I mean it tells me that X is a perfect square but a perfect square could be odd or even right it could be 16 it could be 9 16s and even perfect square and 9 is is an odd one so this doesn't give me much information by itself let's see what statement 2 tells us statement 2 says that Y is equal to Y is equal to Z minus 1 squared so this statement by itself is kind of like statement 1 it just tells me that Y is a perfect square of some integer right because Z is an integer so Z minus 1 is integer so just tells me why is it perfect square and a perfect square could be even or odd but if we take both of these together then something interesting happens for example if we assume that Z is odd then its square will also be odd right 3 squared is 9 7 squared is 49 so if Z is odd then X is odd and then Z minus 1 would be even and then Y would be even so there would be one would be odd one would be even and you could do the other way you could say if Z is even X is even and if Z is even then Z minus 1 is odd and Y is odd and I can prove it to you mathematically let's let's write this with and substitute for Z so X minus y becomes Z squared minus Z minus 1 squared and so this becomes Z squared minus Z squared minus 2z plus 1 and that equals z squared minus z squared plus 2z right distribute the negative sign plus 2z minus 1 the z squares cancel out and we're left with two Z minus 1 if we use both statements X minus y simplifies to 2 Z minus 1 they told us that Z is a positive integer so this this part of the statement right here has to be even it's a multiple of 2 so this is even and if you subtract one from an even number this whole expression has to be odd so both statements together are sufficient to say that X minus y is odd next problem this one looks looks hairy 150 Henry purchased three items during a sale he received a 20% discount off the regular price off the most expensive item and a 10% discount off the regular price of the other two what was the total amount was a total amount of the three discounts greater than 15% of the sum of the regular price okay so what was the total amount of the three discounts it was 20% times let's call X let's just call it item one 20% times item one plus 10% plus 10% times item 2 plus 10% times items item 3 and I'm sorry 10% times item three and I'm assuming this is the most expensive let's just say the second most I guess this is third most expensive maybe they're the same price I don't know and they're asking whether this is the total discount this isn't what he paid this is the discount item one the discounted item two I'm not saying what they actually paid for him to so what the question is was a total amount of the three discounts that's this number greater than greater than 15% of the sum of the regular prices so they're saying was that greater than 15% of i1 plus i2 plus i3 and I think we can simplify this because if we distribute this right hand side you get 0.15 times i1 plus point 0.15 times i2 plus 0.15 times i3 and let's see if we subtract out 0.15 I 1 from both sides you get point C 0.2 minus 0.15 is 0.05 i1 and I want to keep everything positive so let me subtract these from the right hand side so I'm going to track point 1 i2 from so I'm gonna subtract these from both sides of the equation so that's going to be greater than 0.15 I 2 minus 0.1 I 2 so that's point O 5 I 2 plus and I'll do the same thing for I 3 point one five point one so I'm gonna subtract 0.1 from point one five so 0.05 I three all I did going from this to this is I just distributed the point one five and I subtracted and added to simplify a little bit and actually this is interesting too because I just have these 0.05 everywhere that's a positive number and I was able to do that because I just added and subtracted from both sides but if I multiply or divide by a positive number then I don't have to change the inequality and you could say let's just divide both side of this equation by 0.05 or the equivalent is to multiply both sides by 20 but then we're left with I 1 is greater than I 2 plus I 3 and I think this simplification without having read the statements was worth it because we went from something very convoluted in a very convoluted problem statement to something very simple so essentially they're asking us was the price of I 1 was the price of item 1 greater than the price of item 2 plus item 3 this is if we can answer this question we can answer the harder question so statement number 1 the regular price of the most expensive item was $50 I 1 was equal to $50 and the regular price of the next most expensive item was $20 so I 2 was equal to 20 so now the question boils down was 50 greater than 20 plus the third most expensive item well I don't know depends well actually this answers our question right because I was about to say well the third most expensive item maybe it's $30 but by definition we know that it's not $30 why because it was the third most expensive item the second most expensive item is $20 so this thing has to be less than $20 if we are to consider it the third most expensive item so if this thing is less than $20 then the right hand side of the equation is definitely it's going to be actually be less than $40 so this is definitely going to be less than 50 so this is going to be true so it turns out that statement 1 by itself is sufficient because you just have to realize that I 3 has to be less than 20 if I 2 is equal to 20 statement 2 the regular price of the least expensive item was $15 so let's see that statement by itself we get I 1 is greater than I 2 plus 15 where this is the least expensive item so just looking at this we know this is going to be 15 this is going to be more than 15 yeah this is hard you can't say anything about this because maybe I want is maybe I 1 is 17 and is 17 greater than and maybe I 2 is 16 plus 15 and in this case it would not be the case or maybe I 1 is $170 and I 2 is 16 plus 15 in which case it would be the case so statement 2 by itself isn't sufficient so the answer to this is a statement 1 alone is sufficient to answer this question see you in the next video