sambam wrote:
Given that w = |x| and x = 2^b – (8^30 + 8^5), which of the following values for b yields the lowest value for w?
(A) 35
(B) 90
(C) 91
(D) 95
(E) 105
This question has a few noteworthy points.
To get the smallest value of w (which is non negative), 2^b should be as close as possible to \((8^{30} + 8^5)\).
\(8^{30} + 8^5) = (2^{90} + 2^{15})\)
Now a valid question is this: what is closer to \((2^{90} + 2^{15})\): \(2^{90}\) or \(2^{91}\) or higher powers? Let's focus on \(2^{90}\) and \(2^{91}\) only first.
Note a few things: \(2^{91} = 2^{90} * 2^1\)
In other words, it is two times \(2^{90}\) i.e. \(2^{90} + 2^{90}\)
So the question comes down to this: Is \((2^{90} + 2^{15})\) closer to \(2^{90} + 0\) or \(2^{90} + 2^{90}\)
Now, it is obvious that \(2^{15}\) will be much smaller than \(2^{90}\).
\(2^{15}\) is equidistant from 0 and \(2^{16}\) on the number line (because using the same logic, \(2^{16} = 2^{15} + 2^{15}\)).
So \(2^{15}\) will be much closer to 0 compared with \(2^{90}\).
So \((2^{90} + 2^{15})\) is closer to \(2^{90} + 0\) i.e. \(2^{90}\).
Hence, b must be 90.
Answer (B)
_________________
Karishma
Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >