Help! Please!
Prove that if n is an element of Z(the integers) and log2n is rational, then log2n is an integer.
(We can assume that n>1).
Suppose that $\displaystyle \log_2 n = p/q$ where $\displaystyle p/q$ is a positive rational number. This means $\displaystyle 2^{p/q} = n\implies \left( 2^{p/q} \right)^q = n^q\implies 2^p = n^q$. Now the fundamental theorem of arithmetic allows us to factorize $\displaystyle n$, the thing is that all factors much be $\displaystyle 2$'s because the LHS is made out of two, thus, $\displaystyle n=2^m$. This means, $\displaystyle 2^p = 2^{mq}$. Thus, $\displaystyle p=mq$ which means $\displaystyle q|p$ which tells us that $\displaystyle p/q$ is an integer.