Proposition: Show that for an integer $n\geq 2$, the period of the decimal expression for the rational number $\frac{1}{n}$ is at most $n-1$. I'm unsure of where to start. This is my first class on proofs. Do I state:
$\frac{1}{n}=a_n=a_1a_2…a_nb_1b_2…b_n$ with $b$ referring to the repeating part of the expression. I've looked at several other examples but am more confused than aided. I'm unsure how to prove the $n-1$ part. Any help would be appreciated.
Best Answer
When you carry out long division of $1$ by $n$, either the process terminates and you have a finite decimal, or you obtain a sequence of remainders among $1, 2, \dots, n-1$. Once a remainder is repeated, the decimals must start repeating too. Since there are only $n-1$ possible remainders, the repetition must occur by the $n$th decimal place at the latest. The period is then the distance between this and the previous occurrence of the same remainder, which must be at most $n-1$ decimal places.