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we're on problem 107 107 any decimal that has only a finite number of nonzero digits is a terminating decimal for example 20 for the example to give us 24 0.8 2 & 5 0.096 so they're saying if you just have a finite number of these nonzero things at some point you have to end and have zeros just left over so that's what they mean by a terminating decimal if R and s are positive integers so R and s are positive integers so positive integers and the ratio R divided by s is expressed as a decimal so R divided by s so that is expressed as a decimal is R divided by a satyr monnaie tting decimal r divided by s terminating so essentially it wouldn't work if R and n this is 1/3 that's not going to work you're gonna have to point three forever so they're saying does it at some point end this is interesting so statement one tells us that 90 is less than R which is less than 100 that statement by itself doesn't help me now let's say that our was I don't know let's say that our was let me think of a of a good are 90 let's say R was 93 if R is equal to 93 and then s is equal to well let's just say that s for the sake of argument is 3 times R right actually I don't I don't have to pick an R whatever R is I could just pick an S to be 3 times R right and I'm not going to get a terminating decimal on the other hand if I pick s to be 2 times R then I'm gonna have a terminating decimal so statement 1 by itself isn't going to help me right statement number two I hope you understand that I can pick any s to make it terminating or not at this point no matter whether you know R is constrained to between 90 and statement 2 tells us statement 2 tells us that S is equal to 4 B s is equal to 4 B now this is interesting to me so where did this B come from all of a sudden any decimal that has only a finite number of nonzero digits is a terminating decimal for he's okay if R and s arcs are integers and the ratio s is equal to 4 B where did this come from let me look at what is what is I think this is a typo because I'm where this is where did B come from all of a sudden oh yeah when I look at the solution they say S is equal to 4 so that's a that's definitely a typo for B s is equal to 4 okay that makes a lot more sense all right so if s is equal to 4 pretty much any number when you divide any number by 4 you're going to have a terminating decimal it's either going to be divisible by 4 or it's going to be some you know the worst case if you have 1/4 that equals 0.25 so no matter what when you divide something by 4 you're always going to have a terminating decimal it's never going to repeat forever I mean you can try it out with every every you know no matter which of these no matter what number you you divide if you view it as a fraction or kind of your fourth grade remainder problems the remainder is either going to be 1 2 or 3 if the remainder is 1 the decimal is going to be 0.25 if the Rangers - the decimals going to be 0.5 and if the remainder is 3 the decimals going to be 0.75 in any case as long as s equals 4 you have a terminating decimal so statement 2 alone is sufficient and statement 1 really doesn't tell us much and I'm annoyed that they had that B typo there next problem 108 okay they've drawn a triangle they've drawn a triangle right there let me see if I can draw it properly okay then it looks looks like huh all right there you go and let me label it so this is B that's a that's C this is y degrees and this is D and this is X degrees in the figure above what is the value of X plus y so X degrees plus y degrees so x plus y is equal to what hmm all right question number one so they don't tell us anything else they don't tell us this is an equilateral triangle they don't tell us that these angles are equal they don't tell us that symmetric they don't tell us anything about this triangle so far so statement number one tells us that X is equal to 70 well just by X equaling 70 that still doesn't help me I still don't know what Y can be I mean you could draw this interior triangle in a bunch of different ways you can see that the angle could change depending on how much you lift up D or compress D this Y angle can change irrespective of what X is so X is equal to 70 does not in any way give me any information on what Y is statement to statement to triangle a triangle a b c and triangle a DC are both isosceles isosceles so isosceles means that these two sides well actually doesn't mean necessarily that those of those two sides are the same I mean that that's kind of tends to be someone's immediate assumption we say a saucier so maybe that side is equal to that side but no it just means that two of the sides are the same so likewise on this triangle although you can at least here can I make an argument now you can't make an argument which two sides are going to be the same so even if you assumed that both those sides are the same you're still not going to because you don't know you know my gut instinct would say oh that size the same as that side but no I can't make that assumption because it could be that this side is you this side that would still make it isosceles in this and this side is to kind of use like the base side it's a very big difference because if these are the two equal sides and if this is X this is going to be X but then if I take the other scenario where if these are two sides and if this is X then both of these are going to be 180 minus X divided by 2 right because this would be you know the bottom line is this still doesn't get me anywhere even if I know that this is 70 degrees so I'm gonna go with E not enough information to solve this problem 109 109 our positive integers P and Q both greater than n so P Q greater than n and they're positive energy they already told us that positive integers okay they say P minus Q is greater than n P minus Q is greater than n well I'll give you a case right now or that doesn't satisfy this what if what if P is equal to 10 Q is equal to 1 then then we you know and n is equal to 2 right then 10 minus 1 is 9 is definitely greater than 2 but look Q is not greater than 2 I 1 is not greater than 2 so Q wouldn't be greater than n or I could say you know it could be I don't know 110 and 108 would equal and when you subtract you get 2 and in this case Q is bigger than 2 so this doesn't give me enough information I can think of two combinations or two combinations of p qs and ends one that says that q is not greater than n and 1 that says that Q is greater than n so 1 by itself isn't enough statement to statement to Q Q is greater than P Q is greater than P that still doesn't help me and even if I used I both of these both of these statements actually cannot be true simultaneously because if these are both positive integers which they told us right if they're both positive integers well greater than n both greater than M they didn't tell us necessarily that n is positive so n doesn't have to be positive my logic furyk statement 1 still holds but let's see if Q is greater than P if Q is greater than P if this by itself doesn't help me much because I can think of a similar combination to this right I could say Q I could say Q you know Q is equal to 10 P is equal to 9 and I could make n is equal to 2 or I could say n is equal to and is equal to 15 right this in no way constrains what n is so it by itself doesn't solve it but what if we were to use both constraints this is the interesting case what if we were to say that P minus Q is greater than n and Q is greater than P so we know that both P and Q are positive they told us that in the problem statement if P and Q are both positive and P and P sorry and Q is greater than P this is going to end up becoming a negative number right these are both positive and this is the larger of the two positive numbers based on the second constraint so this is going to be a negative number and so if a negative number is greater than n then we can be sure we can be positive that both P and Q are greater than n because they told us that P and Q are positive so actually both of these statements imply that n is negative and is negative because we have a negative number here being greater than n and if n is negative then both P and Q are definitely greater than n so both statements combined are sufficient to answer this question and the answer is actually true i'll see you in the next video