Perform the long division for $\frac{1}{13}$. Identify the repeating block of digits. Does it show cyclic properties if you evaluate $\frac{2}{13}$? Now compute $\frac{3}{13}$, $\frac{4}{13}$, etc. What do you notice? [Page No. 62]

$\frac{1}{13} = 0.076923076923\dots$

$$=0.\overline{076923}\quad(6\text{-digit repeating block})$$

$$\frac{2}{13} = 0.\overline{153846}$$

$$\frac{3}{13} = 0.\overline{230769}$$

$$\frac{4}{13} = 0.\overline{307692}$$

$$\frac{5}{13} = 0.\overline{384615}$$

$$\frac{6}{13} = 0.\overline{461538}$$

$$\frac{7}{13} = 0.\overline{538461}$$

$$\frac{8}{13} = 0.\overline{615384}$$

$$\frac{9}{13} = 0.\overline{692307}$$

$$\frac{10}{13} = 0.\overline{769230}$$

$$\frac{11}{13} = 0.\overline{846153}$$

$$\frac{12}{13} = 0.\overline{923076}$$

Observation: The repeating blocks fall into two cyclic groups:

Group 1 (076923 family):

$$\frac{1}{13}, \frac{3}{13}, \frac{4}{13}, \frac{9}{13}, \frac{10}{13}, \frac{12}{13}$$

Group 2 (153846 family):

$$\frac{2}{13}, \frac{5}{13}, \frac{6}{13}, \frac{7}{13}, \frac{8}{13}, \frac{11}{13}$$

Within each group, the digits are cyclic permutations of one another.