I was asked to prove the following statements:
A rational number $m/n$, where $\gcd (m,n) = 1$, has a pure repeating decimal representation iff the descomposition of $n$ does NOT contain $2$ or $5$ as prime factors.
A rational number $m/n$, where $\gcd (m,n) = 1$, has a mixed repeating decimal representation iff the prime decomposition of $n$ contains $2$ or $5$ and another prime factor
I don't know how to begin. To answer, you don't have to be strict.
Thanks in advance
The first should be a terminating decimal. Note that if $n$ is as specified, it is a factor of some power of $10$. I think what you mean by a pure repeating decimal is one that starts its repeat immediately after the decimal point. Note that $\frac 17=0.\overline{142857}$ is a pure repeating decimal and $7$ has a prime factor other than $2$ and $5$. The statement should be that you get a pure repeating decimal if $n$ has no factors of $2$ or $5$. For the second, note that you can multiply $n$ by the right number of $2$s or $5$s to get a number of the form $10^kp$ where $p$ has no factors of $2$ or $5$. Now use the fact that a fraction with denominator $p$ has a pure repeating decimal.