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Hi, there. I'm happy to give my two cents with this.

Prompt:
x is a positive number
(9^x)+(9^(x+1))+(9^(x+2))+(9^(x+3))+(9^(x+4))+(9^(x+5))=y, Is y divisible by 5?

I totally agree with your approach. . . .

y = (9^x)*(1+9^1+9^2+9^3+9^4+9^5)

The parenthesis indeed is a multiple of 10, so must be divisible by 5. If x = 0 or a positive integer, than y will definitely be divisible by 5.

IF x is a positive rational number, y will not be a whole number. If x is a positive irrational number, for example a logarithm, then y could be either a non-integer or even an integer not divisible by 5 --- (for example, if x = log(17/5)/log(9) --- but that's miles and miles beyond what you need to know for the GMAT.)

I am deeply suspicious of this question, because the "catch" in it seems to depend on an understanding of logarithms, which again is way way beyond the realm of GMAT math. I think someone was trying too hard to write a challenging DS question, and lost all track of what math is actually tested on the GMAT.

So, in short, I agree with you that this is a flawed DS question.

Here's a much more GMAT-like DS question involving laws of exponents.
http://gmat.magoosh.com/questions/1003
When you submit your answer to that question, the following page will have a complete video explanation. At Magoosh, each of our over 800+ questions has its own video explanation. If you are looking for high quality GMAT prep material at a surprisingly affordable price, now is particular good time to check out Magoosh, because we are having a sale that ends later this week.

Does everything I've said make sense? Let me know if you have any further questions.

Mike

_________________
Magoosh GMAT Instructor
http://gmat.magoosh.com/

Prompt: If x and y are integers and xy ≠ 0, what is the value of [x^(-2y)]/[y^(2x)]?

Well, that expression simplifies a little to

[x^(-2y)]/[y^(2x)] = 1/[x^(2y)*y^(2x)] = 1/[x^y*y^x]^2

Statement #2: xy = y/x

Well, with this expression, first multiply by x to clear the fraction, and we get:

(x^2)*y = y

Since we know y does not equal zero, we can divide by y, which results in:
x^2 = 1
x = +/-1
(Always remember you need to include the +/- when you take the squareroot of a variable.)

Thus, we know x = +/-1 and we have absolutely no idea what y is --- y could be absolutely an integer other than zero. That's why statement #2 is insufficient.

Does this make sense? Let me know if you have any further questions.

Mike

_________________
Magoosh GMAT Instructor
http://gmat.magoosh.com/