Question 161
$p$ is a prime number such that the period of its decimal reciprocal is $200$. That is,
$$
\frac{1}{p}=0 . X X X X \ldots
$$
for some block of $200$ digits $X$, but
$$
\frac{1}{p} \neq 0 . Y Y Y Y \ldots
$$
for all blocks $Y$ with less than $200$ digits. Find the $101$st digit, counting from the left, of $X$.
$$
\frac{1}{p}=0 . X X X X \ldots
$$
for some block of $200$ digits $X$, but
$$
\frac{1}{p} \neq 0 . Y Y Y Y \ldots
$$
for all blocks $Y$ with less than $200$ digits. Find the $101$st digit, counting from the left, of $X$.