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# GMAT: Data sufficiency 31

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

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• What about question 126? What if n is a negative number?
• 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)
• 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)
• 6/15 is not a valid example because 15 is a factor of 15, but not a factor of 6.
(1 vote)