[Periodic and Non-periodic Fractions]

The number $N=142857$ has several other properties. For example: $2 \cdot 142857=285714, 3 \cdot 142857=428571$, ..., that is, when multiplied by 1, 2, 3, ..., 6, the digits are cyclically permuted; $14+28+57=99 ; N^{2}=20408122449$, $20408+122449=142857=N$.

Similar operations can be performed with other periods of fractions. What happens with the numbers $1 / 17$, $1 / 19$? Explain these facts.
When decomposing $1 / 7$ into a decimal fraction, the sequence of remainders is arranged as follows: $r_{0}=1, r_{1}$ $=3, r_{2}=2, r_{3}=6, r_{4}=4, r_{5}=5, r_{6}=1, \ldots$

The first property is explained by the equalities $2 \cdot r_{0 / 7}=r_{2 / 7}, 3 \cdot r_{0 / 7}=r_{1 / 7}, 4 \cdot r_{0 / 7}=r_{4 / 7}, \ldots$

The explanation of the second property is obtained by transitioning to the decimal representation in the equality $1 / 7+2 / 7+4 / 7=1$. To explain the last property, write $N$ as $\frac{10^{6}-1}{7}$. From this, $N^{2}=\frac{\left(10^{6}-1\right)^{2}}{49}$. The number obtained by adding the halves of the number $N^{2}$ will be the period of the fraction

$\sum_{k=1}^{\infty} \frac{N^{2}}{10^{6 k}}=N^{2} \cdot \frac{1}{10^{6}-1}=\frac{10^{6}-1}{49}=\frac{142857}{7}$

Since $\left\{{ }^{142857} /{ }_{7}\right\}=1 / 7=0,(142857)$, the number obtained from the halves of $N^{2}$ will be $N$.