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alright we're on problem 125 on page 288 125 if R and s are positive integers is R over s an integer is R over as an integer so that's really just another way of saying is s divisible into R so let's see what the statements are statement every factor of S is also a factor of R so that's well that that's that answers our question every factor of S is fact factor of R well let me ask the question what is the largest factor of s well the largest factor of S is s right so this statement tells us that since s is a factor of s that s is also going to be a factor of R and something being a factor of something means that it's divisible into it so that means that s is divisible into R so this means that our / s is an integer so it answers our question so statement 1 alone is sufficient statement 2 tells us every prime factor of S is also a prime factor of R every prime factor so let me think of know immediately I can think of a case where that doesn't hold up that where I could have every prime factor being a factor of R so let's say s is equal to 4 and its prime factorization is 2 times 2 and let's say that R is equal to 6 and its prime factorization is 2 times 3 so we have a case here where every factor every prime factor of s is a prime factor of R right the only prime factor of S is 2 and that's a prime factor of R but if we were to say R over s R over s would be 6 over 4 which is not an integer so even though these satisfy the second conditions this is an integer but then I could have instead of making R equal to 6 I could have made R equal to 4 I'm sorry I could have made R is equal to 8 which is equal to two times two times two and in this case it would have been an integer R over s would be equal to 8 over 4 which is equal to 2 so statement 2 really doesn't give us information as to whether R / s is an integer so statement 1 alone is sufficient next problem switch colors 126 126 if Z to the N is equal to 1 z to the N is equal to 1 what is the value of Z Z equals what so statement 1 tells us N is a nonzero integer and does not equal to 0 so let's think about this for something to some power to be equal to 1 what do we know about it well if the if anything to the zeroth power is equal to 1 right anything that the 0th power is equal to 1 but they just told us that it's nonzero so we can't use this condition so that actually does restrict see a good bit so if N is a nonzero number what can I raise to the power to equal 1 well clearly 1 right 1 2 anything is going to be equal to 1 but what other well there's also the possibility of negative numbers right negative 1 if to the nth power is equal to 1 if n is even if n is even right so and I think these are the only two if we take imaginary numbers out of the picture and I think on the GMAT we assume that we don't have any imaginary numbers so assuming that the only if if we know that n does not equal 0 and Z to the N is equal to 1 the only possibilities that this allows for is that Z is equal to 1 or negative 1 and if it's negative 1 then n would have to be an even number but they didn't restrict that yet so statement 1 by itself it helps us but it doesn't actually give us enough information we can just narrow it down to Z being 1 or negative 1 let's see what statement 2 tells us statement 2 tells us Z is greater than 0 so that by itself is useless information because we if Z is greater than zero and n can be zero right listen we're assuming we don't have statement one yet so if C is greater than zero and n could be zero Z could be a hundred to the 0th power right as you could be a hundred and that equals one is he could be 99 to the zero to the power and that could be equal one so Z could be anything to the zeroth power as long as it's greater than zero so this by itself doesn't help us but if we use statement to statement one in conjunction then it's interesting because statement one essentially told us that Z has to be one or negative one statement 2 tells us Z has to be greater than zero so if you use both the conditions combined it forces us to say well then Z has to be equal to positive one because negative 1 is not greater than zero so both statements combined are sufficient to answer this question next problem one 27:27 okay so they've written this thing they write s is equal to 2 over n all of that over 1 over X plus 2 over 3 X and they say in the expression above if xn does not equal 0 xn does not equal 0 so essentially saying that neither X nor n is 0 say what is the value of s S is equal to what so even before looking the statement that it's just I just want to simplify this just because it's too complex right now so let's see this is equal to 2 over N over let's see if common denominator 3 x + C this is 3 over 3 X s same thing as 1 over X plus 2 so that equals 2 over n over 5 over 3 X which is equal to 2 over N times 3 x over 5 which is equal to 6 x over 5 n that's a much more pleasing thing to look at and try to get your your brain around so statement 1 tells us X is equal to 2 n X is equal to 2 n so if X is equal to 2n let's just substitute that into this statement for s so then s would be equal to 6 times X which we now is equal to 2 n divided by 5n and cancel out so it equals 12 over 5 so statement 1 alone is sufficient to answer this question statement 2 tells us that n is equal to 1/2 well that's fairly useless because this is what we simplify it s down to if we put 1/2 here then we get s is equal to 6x over 5 times 1/2 so that's 5 halves but yeah I could simplify this more but we still don't know what X is you can't cancel the XS out or anything so this still doesn't this alone doesn't help you at all so the answer is statement 1 alone is sufficient and statement 2 by itself is fairly useless problem 128 128 if X is an integer is x times I think that says the absolute it's kind of strange to look at it like that but they're saying is x times the absolute value of X less than 2x and they tell us that X is an integer okay statement 1 tells us that X is less than 0 so how does that help us if X is less than 0 then this on the right-hand side is going to be less than 0 and this will also be less than 0 because you'll have a negative number times a positive number it will both be less than 0 but if we say if we pick a if we make X is equal to negative 1 then this would not then this then this will not be true because you'll have negative 1 times 1 so you'll have 1 is less than negative 2 sorry you'd have negative 1 times 1 which is negative 1 which is less than 2 which is not true it's not less than 2 or if X was if X was less than negative 2 if X was negative 3 then you would have negative 3 times 3 right because the absolute value of negative 3 so you would have minus na is less than minus three times two minus six this is true so just this condition alone doesn't get us there because I can still pick an X that meets this this condition and depending on whether I make that X greater than or less than negative two I can make this true or not so statement 1 by itself isn't enough statement 2 tells us X is equal to minus 10 well this is an easy one we can just test it so if we have minus 10 times the absolute value of minus 10 and we're in test whether that's less than 2 times minus 10 so the absolute value of minus this is minus 10 times positive 10 so it's minus 100 which is less than minus 20 which is completely true so statement two alone is sufficient to answer this question and I'm almost out of time so I'll see you in the next video