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GMAT: Data sufficiency 31

125-128, pg. 288. Created by Sal Khan.

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  • blobby green style avatar for user Garry Vardanyan
    What about question 126? What if n is a negative number?
    (3 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      The question is whether we can tell the value of Z, given that Z ⁿ = 1

      If n = 0, then we cannot tell the value of Z. Any number to the 0 power will result in a value of 1 (except if Z = 0, when the result is undefined), so condition 1 alone is not sufficient.

      So you are asking about the case where n is a negative number. If n is a negative number, then we are talking about a reciprocal power of Z.
      As an example, if Z = 4 and n = -2, then Z ⁿ = 4 ⁻² = (¼ )² = 1/16 → That doesn't equal 1, so it is not useful to us. Try a few of those and it can seem hopeless. So when would Z ⁿ = 1 ?

      We get back to the central argument that Sal made. The only number raised to the n power that ALWAYS equals 1 is 1

      And that is true whether n is positive or negative.
      The other number raised to the n power that SOMETIMES equals 1 is -1
      If n is an even integer *whether negative or positive,* the result will be 1.
      Let's let n = -2 again:
      Z ⁿ = 1
      Z ⁻² = ?
      If Z = 1, 1 ⁻² = 1/1² = 1/1 = 1
      If Z = - 1, (-1) ⁻² = 1/((-1)²) = 1/1 = 1
      Of course if Z = -1 and n = -3, then Z ⁻³ = (-1)⁻³ = 1/((-1)⁻³) = 1/(-1) = -1
      That breaks the condition in the prompt that says that Z ⁿ = 1
      Since Z ⁿ must equal 1, we can eliminate negative values of n that are odd.
      So, we are left still not knowing the value of Z, because we have two different alternatives. It can either be 1 or -1, which is not good enough.

      Condition 2 solves it for us, because it says that Z > 0. Now there is only one possible value for Z. Both statements are required, although neither is sufficient alone.
      (1 vote)
  • blobby green style avatar for user Virat
    I have a doubt with question 125. "statement 1" says, "every factor of s is a factor of r". So for example, say r/s is 6/15.. so 6 represents r and 15 represents s.... now according to statement 1, every factor of s (15 in this case) - say 3 is a factor, is a factor of r(6).. but r/s is not an integer. So how is statement 1 sufficient ? Please clarify
    (1 vote)
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Video transcript

All right, we're on problem 125 on page 288. If r and s are positive integers, is r/s an integer? Is r/s 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 1. Every factor of s is also a factor of r. That answers our question. Every factor of s is factor of r. Well, let me ask you a question. What is the largest factor of s? The largest factor of s is s. 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 r/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-- Immediately, I can think of a case where that doesn't hold up. 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. The only prime factor of s is 2. And that's a prime factor of R. But if we were to say R/s, R/s would be 6/4, which is not an integer. So even though these satisfy the second condition, this isn't an integer. But then I could have-- instead of making R equal to 6, I could've made R equal to 4. Sorry, I could have made R is equal to 8, which is equal to 2 times 2 times 2. And in this case, it would have been an integer. R/s would be equal to 8/4, 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. If 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. n 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 anything to 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 z 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 to anything is going to be equal to 1. But what other? Well, there's also the possibility of negative numbers, right? Negative 1 to the nth power is equal to 1, if n is even. 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 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 if z is greater than 0 and n can be 0, right? We're assuming we don't have statement 1 yet. So if z is greater than 0 and n could be 0, z could be a hundred to the 0th power. z could be a hundred and that equals 1. z could be 99 to the 0th power, and that could be equal to 1. So z could be anything to the 0th power as long as it's greater than 0. So this, by itself, doesn't help us. But if we use statement 2 and statement 1 in conjunction, then it's interesting. Because statement 1 essentially told us that z has to be 1 or negative 1. Statement 2 tells us, z has to be greater than 0. So if you use both the conditions combined, it forces us to say, well, then z has to be equal to positive 1. Because negative 1 is not greater than 0. So both statements combined are sufficient to answer this question. Next problem. 127. OK, so they've written this thing. They write, s is equal 2/n, all of that over 1/x plus 2/3x. And they say in the expression above, if xn does not equal 0-- so essentially saying that neither x nor n is 0. What is the value of s? s is equal to what? So even before looking at the statements, I just want to simplify this. Just because it's too complex right now. So let's see. This is equal to 2/n over-- let's see. You have a common denominator 3x. Let's see. This is 3 over 3x. That's the same thing as 1/x. Plus 2. So that equals 2/n over 5/3x, which is equal to 2/n times 3x/5. Which is equal to 6x/5n. That's a much more pleasing thing to look at and try to get your brain around. So statement 1 tells us, x is equal to 2n. So if x is equal to 2n, let's just substitute that into this statement for s. Then, s would be equal to 6 times x, which we know is equal to 2n. Divided by 5n. n's cancel out. So it equals 12/5. 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 simplified 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 5/2. I can simplify this more, but we still don't know what x is. You can't cancel the x's out or anything. 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. 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. OK, 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. They'll both be less than 0. If we make x is equal to negative 1, then this would not-- 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 less than negative 2, if x was negative 3, then you would have negative 3 times 3. Because the absolute value of negative 3. So you would have minus 9 is less than minus 3 times 2. Minus 6. This is true. So just this condition, alone, doesn't get us there. Because I can still pick an x that meets this condition. And depending on whether I make that x greater than or less than negative 2, 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 going to test whether that's less than 2 times minus 10. So the absolute value of-- So this is minus 10 times positive 10. So it's minus 100, which is less than minus 20. Which is completely true. So statement 2, alone, is sufficient to answer this question. And I'm almost out of time, so I'll see you in the next video.