QUESTION 2
Easy
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?
SOLUTION
Long division shown in figure.
\(\frac{1}{13} = 0.0769230769230... = 0.\overline{076923} \)
Here, the repeating block is 076923.
Similarly, \(\frac{2}{13} = 0.\overline{153846} \)
The repeating block is 153846.
\(\frac{3}{13} = 0.\overline{230769} \)
\(\frac{4}{13} = 0.\overline{307692} \)
\(\frac{5}{13} = 0.\overline{385615} \)
\(\frac{6}{13} = 0.\overline{461538} \)
\(\frac{7}{13} = 0.\overline{583461} \)
Here, we notice that the repeating blocks fall into two cyclic groups:
\(\frac{1}{13},\frac{3}{13},\frac{4}{13}\) sharing the (076923) group while \(\frac{2}{13},\frac{5}{13},\frac{6}{13},\frac{7}{13}\) share the (153846) group.
\(\frac{1}{13} = 0.0769230769230... = 0.\overline{076923} \)
Here, the repeating block is 076923.
Similarly, \(\frac{2}{13} = 0.\overline{153846} \)
The repeating block is 153846.
\(\frac{3}{13} = 0.\overline{230769} \)
\(\frac{4}{13} = 0.\overline{307692} \)
\(\frac{5}{13} = 0.\overline{385615} \)
\(\frac{6}{13} = 0.\overline{461538} \)
\(\frac{7}{13} = 0.\overline{583461} \)
Here, we notice that the repeating blocks fall into two cyclic groups:
\(\frac{1}{13},\frac{3}{13},\frac{4}{13}\) sharing the (076923) group while \(\frac{2}{13},\frac{5}{13},\frac{6}{13},\frac{7}{13}\) share the (153846) group.