Problem Solving Pack 4, Question 3) If a, b and c are consecutive...

Hi All,

To start, the answer choices to this question are rather 'spread out', so we might be able to use that spread to our advantage (and avoid some calculations).

We're told that a, b and c are consecutive positive integers and a < b < c. We're asked for the MINIMUM value of \(3^{2bc}\) / \(3^{2ab}\)...

Since the prompt asks for the MINIMUM value, it's likely that we'll have to make the 3 variables as small as possible, but we'll have to check to make sure that that's what is required to get to the correct answer. The smallest 3 numbers that 'fit' are...

a = 1
b = 2
c = 3

This makes the calculation...

\(3^{12}\) / \(3^{4}\) = \(3^{8}\)

\(3^{8}\)= \(9^{4}\) = \(81^{2}\) = ABOUT \(80^{2}\) = ABOUT 6400, so the answer appears to be D. We just need to confirm that that's the case....

IF....
a = 2
b = 3
c = 4

This makes the calculation...

\(3^{24}\) / \(3^{12}\) = \(3^{12}\)

\(3^{12}\) is clearly bigger than \(3^{8}\), meaning that increasing the values of a, b and c will lead to a LARGER end result. This proves that D is actually the smallest possible result.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

since a, b, and c are consecutive integers, and positive: we can rewrite the fraction as

3^[2bc-2ab] or
3^[2(bc-ab)] or
3^[2(b(c-a))]

since a,b,c are consecutive integers, c-a is always = 2 regardless of the values they take.

now, we are left with 3^4b
to minimize the value, we need to take the minimal value of b, while taking into consideration that a,b,c - must be positive and numbers are consecutive numbers.
minimal value of b is 2.

now, 3^8 = 9^4 or 81^2.
don't need to solve for the exact value. the correct answer choice must have 1 at the end, and since A cannot be the answer, the only correct answer that it can be is D.

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