Bunuel wrote:
Is p an odd integer?
(1) \((p^5)^5\) is an odd integer.
(2) \(\sqrt[5]{\sqrt[5]{p}}\) is an odd integer.
Target question: Is p an odd integer? Statement 1: \((p^5)^5\) is an odd integer. Since \((p^5)^5 = p^{25}\), statement 1 is really telling us that \(p^{25}\) is an odd integer.
There are several values of p that satisfy statement 1. Here are two:
Case a: If \(p = 1\), then \(p^{25} = 1^{25} = 1\), and \(1\) is an odd integer. Since \(p = 1\), the answer to the target question is
YES, p is an odd integerCase b: If \(p = \sqrt[25]{3}\), then \(p^{25} = (\sqrt[25]{3})^{25} = 3\), and \(3\) is an odd integer. Since \(p = \sqrt[25]{3}\), the answer to the target question is
NO, p is not an odd integerSince we can’t answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: \(\sqrt[5]{\sqrt[5]{p}}\) is an odd integer.First off, \(\sqrt[5]{\sqrt[5]{p}} = \sqrt[25]{p}\), which means statement 2 is telling us that \(\sqrt[25]{p} =\) some odd integer
Let's raise both sides of this equation to the power of \(25\) to get: \((\sqrt[25]{p})^{25} =\) (some odd integer)^25
Simplify: p = (some odd integer)^25
Since any odd integer raised to the power of 25 will always be odd, we can be certain that
p is oddStatement 2 is SUFFICIENT
Answer: B