Given that both x and y are positive integers, and that \(y=3^{x–1}–x\), is y divisible by 6?
(1) x is a multiple of 3
Let x = 3k where k is positive integer > 0
\(y=3^{x – 1} – x\) = \(3^{3k – 1} – 3k\)
Since both the entities in y are multiple of 3, y is multiple of 3. It is only when k is even y would be odd and when k is odd y would be even.
If k= 1
\(y=3^{3.1 – 1} – 3.1\) = \(y=3^2 – 3\) = 6 YES.
If x = 2
\(y=3^{3.2 – 1} – 3.2\) = \(y=3^5 – 6\) = 237 NO.
INSUFFICIENT.
(2) x is a multiple of 4
Let x = 4k where k is positive integer > 0
\(y=3^{x–1}–x\) = \(3^{4k – 1} – 4k\)
Since \(y=3^{4k – 1}\) would always be odd and 4k would always be even, y would always be odd, a non multiple of 6
though \(y = 3^{4k – 1} – 4k\) would be multiple of 3 when k is a multiple of 3.
SUFFICIENT.
ANSWER B.
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