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For how many positive integers n>1 is it true that \(2^{42}\) is a perfect nth power?

 Jan 31, 2022
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For any positive integer to satisfy the conditions, it has to be a perfect power of \(2\), and it should be able to be equivalent to \(2^{42}\), with integer values.

 

The only possible options are:

\(2\), can be written as \(2^{42}\)

\(4\), can be written as \(({2^2)}^{21}\)

\(8\), can be written as \((2^3)^{14}\)

 

Thus... the answer is \(\color{brown}\boxed3\)

 Jan 31, 2022

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