Manhattan GRE's Math Beast Challenge problem for this week

This is a "quantitative comparison" problem. Two expressions are placed in two columns-- Column A and Column B. You have to determine which column contains the greater quantity. Your options are usually:

a. Column A is greater.
b. Column B is greater.
c. Columns A and B are equal.
d. It cannot be determined with the information given.

This week's quantitative comparison:

Column A:

a¹⁰⁰ - 1
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(a⁷² + 1)(a³⁵ + 1)(a¹⁸ + 1)


(Sorry for the bad HTML. The above expression is supposed to be a fraction with a numerator of (a^100 - 1), and a denominator of [(a^72 + 1)(a^35 + 1)(a^18 + 1)].)



Column B:

a¹⁸ + 1

Answer:


(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.



I'm going with (D). Column A includes odd exponents, which is relevant if a happens to be negative, since a negative number taken to an odd power will be negative. We don't know whether a is less than, equal to, or greater than zero. We also don't know whether a is an integer: if, for example, a is a fraction greater than zero and less than one, taking it up to the hundredth power will only shrink the quantity, not augment it. Example: (1/2)^3 = 1/8, which is less than 1/2, whereas 2^3 = 8, with 8 being greater than 2.

Comments?

ADDENDUM: I should also note that, if a = -1, the fraction in Column A will have a denominator of 0 and thus be undefined.



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