## Task 3 - 190813

Given are the four periodic decimal fractions

$$
p=0, \overline{3456} \ldots, \quad q=0,3 \overline{456} \ldots, \quad r=0,34 \overline{56} \ldots, \quad s=0,345 \overline{6}
$$

a) Determine all natural numbers $n$ for which the following statement holds: In the $n$-th decimal place, all four decimal fractions $p, q, r, s$ have the same digit.

b) Determine all natural numbers $m$ for which the following statement holds: In the $m$-th decimal place, no two of the four decimal fractions $p, q, r, s$ have the same digit.
a) For $n=1,2,3 \quad$ (1) the stated assertion holds. If it holds for a natural number

$$
n=4+k \quad \text { with } k \geq 0
$$

it follows: In the $n$-th decimal place, $s$ has the digit 6. This digit appears in $p$ or $q$ at the $(4+k)$-th place only if $k$ is divisible by 4 or 3, respectively. Therefore, $k$ is divisible by the least common multiple (LCM) of 4 and 3, which is 12. Thus, only the numbers

$$
n=4+12 g \quad(g=0,1,2, \ldots)
$$

can have the property in question.

These numbers do have this property; since $12 g$ is divisible by 4, 3, and 2, the digit 6 appears at the $(4+12 g)$-th place in $p, q$, and $r$ just as it does in $s$.

Therefore, exactly the numbers (1), (2) satisfy the condition stated in part a).

b) For $m=1,2,3$ the stated assertion is false. At the fourth decimal place, all decimal fractions $p, q, r, s$ begin a period with the digit 6. As shown in part a), this repeats every 12 places. Therefore, it suffices to check the twelve numbers

$$
m=4,5, \ldots, 15
$$

What is found for any of these values (3) (regarding the property to be investigated) also holds for all values of $m$ that result from the relevant value by adding arbitrary integer multiples of 12. (In the following table, the occurrence of the same digits is marked by framing. 1st column $=\mathrm{m}$-th place in ...)

| $m$ | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $p$ | 6 | 3 | 4 | 5 | 6 | 3 | 4 | 5 | 6 | 3 | 4 | 5 |
| $q$ | 6 | 4 | 5 | 6 | 4 | 5 | 6 | 4 | 5 | 6 | 4 | 5 |
| $r$ | 6 | 5 | 6 | 5 | 6 | 5 | 6 | 5 | 6 | 5 | 6 | 5 |
| $s$ | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |

It turns out that among the numbers (3), exactly $m=5$ has the property in question. Therefore, exactly the numbers

$$
m=5+12 h \quad(h=0,1,2 \ldots)
$$

are the ones sought in part b).