Question 177
a) Consider a prime $p$ that divides $10^{n}+1$ for some positive integer $n$. For example, $p=7$ divides $10^{3}+1$. By analysing the principal period of the decimal representation of $\frac{1}{p}$, verify that the number of times the digit $i$ appears equals the number of times the digit $9-i$ appears, for each $i \in\{0,1,2, \ldots, 9\}$.
b) Consider a prime number $p$ that does not divide 10 and suppose that the period of the decimal representation of $\frac{1}{p}$ has length $2k$. Is it always possible to split the period into two numbers whose sum is $10^{k}-1$? For example, the period of $\frac{1}{7}$ has length $6=2k$, since it is equal to 142857. Note that $142+857=999=10^{3}-1=10^{k}-1$.
c) Let
$$
x=\frac{1}{1998}+\frac{1}{19998}+\frac{1}{199998}+\ldots
$$
When we write $2x$ as a decimal number, what is the $59^{\text{th}}$ digit after the decimal point?
b) Consider a prime number $p$ that does not divide 10 and suppose that the period of the decimal representation of $\frac{1}{p}$ has length $2k$. Is it always possible to split the period into two numbers whose sum is $10^{k}-1$? For example, the period of $\frac{1}{7}$ has length $6=2k$, since it is equal to 142857. Note that $142+857=999=10^{3}-1=10^{k}-1$.
c) Let
$$
x=\frac{1}{1998}+\frac{1}{19998}+\frac{1}{199998}+\ldots
$$
When we write $2x$ as a decimal number, what is the $59^{\text{th}}$ digit after the decimal point?