Question 2

We will use long division to find repeating decimal expansions.

Step 1 — Long division for 1/13

Let's divide 1 by 13. We will perform long division.

The remainder 1 repeats. This means the digits will repeat from this point. The decimal expansion is $\mathbf{0.076923076923\dots}$.

$\boxed{1/13 = 0.\overline{076923}}$

Step 2 — Evaluate 2/13

Let's compute the decimal expansion for 2/13. We use long division.

$2/13 = 0.153846153846\dots$

The repeating block is 153846. This block is different from 076923. Thus, it does not show cyclic properties of the same block as 1/13.

$\boxed{2/13 = 0.\overline{153846}}$

Step 3 — Compute 3/13, 4/13, 5/13, 6/13

We will find the decimal expansions for these fractions.

$3/13 = 0.230769230769\dots$

$\boxed{3/13 = 0.\overline{230769}}$

$4/13 = 0.307692307692\dots$

$\boxed{4/13 = 0.\overline{307692}}$

$5/13 = 0.384615384615\dots$

$\boxed{5/13 = 0.\overline{384615}}$

$6/13 = 0.461538461538\dots$

$\boxed{6/13 = 0.\overline{461538}}$

Step 4 — Observe the patterns

Let's look at all the repeating blocks.

The repeating block for 1/13 is 076923. The repeating block for 2/13 is 153846. The repeating block for 3/13 is 230769. The repeating block for 4/13 is 307692. The repeating block for 5/13 is 384615. The repeating block for 6/13 is 461538.

We can see two distinct sets of digits. Set 1: {0, 7, 6, 9, 2, 3}. Set 2: {1, 5, 3, 8, 4, 6}.

The blocks for 1/13, 3/13, 4/13 are cyclic permutations of Set 1. For example, starting 076923 from '2' gives 230769. Starting 076923 from '3' gives 307692.

The blocks for 2/13, 5/13, 6/13 are cyclic permutations of Set 2. For example, starting 153846 from '3' gives 384615. Starting 153846 from '4' gives 461538.

Answer

(i) The repeating block of digits for $\mathbf{1/13}$ is 076923. (ii) When evaluating $\mathbf{2/13}$, the repeating block is 153846. This block is not a cyclic permutation of the repeating block of $\mathbf{1/13}$. It shows a distinct cyclic property. (iii) The decimal expansions of $\mathbf{1/13}$, $\mathbf{3/13}$, and $\mathbf{4/13}$ use cyclic permutations of the digits {0, 7, 6, 9, 2, 3}. The decimal expansions of $\mathbf{2/13}$, $\mathbf{5/13}$, and $\mathbf{6/13}$ use cyclic permutations of the digits {1, 5, 3, 8, 4, 6}. These two groups exhibit distinct cyclic properties.