Co-prime integer (high level question)

Nothing wrong with how you started the problem. If a = 2009, the fraction is equal to:

[ (a^2 - 9)(a^2 - 1) - 9 ]/ [ (a^2 - 25)(a^2 - 1) + 135 ]

Now if you multiply out the products on the top and bottom, you get:

= [a^4 - 10a^2 + 9 - 9] / [a^4 - 26a^2 + 25 + 135]

= (a^4 - 10a^2) / (a^4 - 26a^2 + 160)

and we can now factor on top and bottom:

(a^2)(a^2 - 10) / (a^2 - 16)(a^2 - 10)

Now one factor cancels, leaving us with:

a^2/(a^2 - 16)

Now, plugging in a = 2009, this is equal to:

2009^2 / 2009^2 - 16 = 2009 / 2013*2005

Neither factor in the denominator could possibly share a divisor with 2009 (they're too close together), so this fraction is completely reduced. So the answer is 2009 + 2013*2005, which is a number in the millions.

This is a really awkward question. There's nothing mathematically interesting about it unless the answer turns out to be something very simple. It's not very satisfying to do all this work only to find that the answer is 4,038,074. Where are these questions from? They aren't GMAT-like at all.

Thanks! it was from my college professor, I was just really stuck on it, even though it is beyond GMAT.
Thank you again for explanation! I completely didn't think of multiplying top and bottom, this is where I was stuck.

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