# Rational doubt ( doubt in rational number) [duplicate]

If there is a prime number x, if we reciprocate it we will get 1/x.

Reciprocal of prime number will be a rational number , Except 1/2 and 1/5 , every number which is reciprocal of prime number is a recurring non terminating decimal number.

Question- How to find that after how many digit recurring numbers will repeat ? How to find the recurring numbers?

## marked as duplicate by MJD, Chris Janjigian, Namaste, colormegone, ml0105Apr 30 '14 at 17:01

• The period $n$ in which $\frac1p$ repeats is the smallest $n$ such that $10^n-1$ is a multiple of $p$. For example, $10^6-1 = 999999$ is a multiple of $7$, but $10^5-1$ is not, so $\frac17$ repeats every 6 digits. However, no general rule is known that describes the smallest such $n$ in general. – MJD Apr 30 '14 at 16:29