OG 13, PS Q 117

[GMATNR]

If n=3^8-2^8, which of the following is NOT a factor of n?
a) 97, b) 65, c) 35, d) 13, e) 5  c

--> How do you solve this in an easy way? E.g. the OG suggests to factor the expression into (3^4+2^4)(3^2+2^2)(3^2-2^2) - how do you easily get there? Is there a better / more intuitive way?

Thanks!

[GMATGuruNY]

If n=3^8-2^8, which of the following is NOT a factor of n?
a) 97, b) 65, c) 35, d) 13, e) 5  c

a² - b² = (a+b)(a-b).

Thus:
3� - 2�

= (3� + 2�)(3� - 2�)

= (81 + 16)(81 - 16)

= (97)(65) = 97*5*13.

The last line indicates that A, B, D and E are all factors of the given expression.

The correct answer is C.

[ceilidh.erickson]

There is another way to at least narrow down the answer choices, if you weren't able to solve.

Since the question asks "which of the following is NOT a factor," we can infer that 4 out of 5 answer choices will be factors.

It would be impossible for the correct answer to be 5. That would mean that 65 and 35 are factors, but 5 itself (a factor of the other two) is not. This is impossible, so the answer cannot be E.

By the same logic, the answer cannot be 13, which is also a factor of 65. If 65 is a factor, then 13 has to be as well. D is out.

If we've eliminated 5 and 13 as answer choices, it means that they MUST be factors. And if 5 and 13 are both factors, then 65 must also be a factor, because 65 = 5 x 13.

Now we've narrowed it down to A or C, and we have a 50/50 shot of guessing correctly.