Let $$ P = \sqrt{\frac{3\times10^{n}}{k}} $$
Find all $k$ positive integers so that $P$ is rational and belongs to $(0,1)$ for all $n$ positive integers.

  • $\begingroup$ You want to find all pairs of positive integers $(k,n)$? Because no single $k$ will work with every $n$. $\endgroup$ – Fimpellizieri Nov 3 at 22:43
  • $\begingroup$ Sorry, I edited. I had misplaced the numerator and denominator. I need to find all k's for all n's. $\endgroup$ – Juan Ernesto Montero Nov 3 at 22:44
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    $\begingroup$ Still no such $k$. $\endgroup$ – Gae. S. Nov 3 at 22:49
  • $\begingroup$ @Gae.S.Why? Can you show me? if $n=1$, $k$ needs to be $30^3$. If so, $P=1/30$ $\endgroup$ – Juan Ernesto Montero Nov 3 at 22:53
  • $\begingroup$ If $n$ is odd then $k=30\cdot t^2$ for any $t>10^{(n-1)/2}$ otherwise if $n$ is even you have $k=3\cdot t^2$ for $t>10^{n/2}$ $\endgroup$ – kingW3 Nov 3 at 22:56

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