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Is \(p\) an odd integer?
(1) \((p^5)^5\) is an odd integer.
(2) \(\sqrt[5]{\sqrt[5]{p}}\) is an odd integer.
(1) \((p^5)^5\) is an odd integer.
(2) \(\sqrt[5]{\sqrt[5]{p}}\) is an odd integer.
06 Apr 2023, 04:46
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Official Solution:
Is \(p\) an odd integer?
(1) \((p^5)^5\) is an odd integer.
The above implies that \(p^{25}\) is an odd integer. If \(p\) itself is an odd integer, it would give a YES answer to the question. However, \(p\) can also be an irrational number, like the 25th root of an odd integer, for example, \(\sqrt[25]{2}\), giving a NO answer to the question. Not sufficient.
(2) \(\sqrt[5]{\sqrt[5]{p}}\) is an odd integer.
The above implies that \(\sqrt[25]{p}\) is an odd integer (\(\sqrt[5]{\sqrt[5]{p}}=\sqrt[5]{p^{\frac{1}{5}}}=p^{\frac{1}{25}}=\sqrt[25]{p}\)). Taking it to the 25th power gives \(p = odd^{25}\). Since an odd number raised to a positive integer power remains odd, then \(p\) must also be an odd integer. Sufficient.
Answer: B
Is \(p\) an odd integer?
(1) \((p^5)^5\) is an odd integer.
The above implies that \(p^{25}\) is an odd integer. If \(p\) itself is an odd integer, it would give a YES answer to the question. However, \(p\) can also be an irrational number, like the 25th root of an odd integer, for example, \(\sqrt[25]{2}\), giving a NO answer to the question. Not sufficient.
(2) \(\sqrt[5]{\sqrt[5]{p}}\) is an odd integer.
The above implies that \(\sqrt[25]{p}\) is an odd integer (\(\sqrt[5]{\sqrt[5]{p}}=\sqrt[5]{p^{\frac{1}{5}}}=p^{\frac{1}{25}}=\sqrt[25]{p}\)). Taking it to the 25th power gives \(p = odd^{25}\). Since an odd number raised to a positive integer power remains odd, then \(p\) must also be an odd integer. Sufficient.
Answer: B
