The length of the repeated sequence of a rational number

Question

I saw somewhere that given p a co-prime to 10, the length of the repeated sequence of $\frac1p$ will be longer than $\log_{10} p$, and I can't find an explanation why.

Any ideas?

Answer 1

If the repeating sequence has $n$ digits then $p$ has to divide $10^n-1$

so $p$ cannot be bigger than $10^n-1$

so $p$ must be less than $10^n$

so $\log_{10}p$ must be less than $n$