## Task $32 / 70$

Determine the smallest natural number $k$ that begins with the digit 7 (when represented in the decimal system) and satisfies the following additional property:

If the leading digit 7 is removed and appended to the end, the newly formed number $z=\frac{1}{3} k$.
We write $k=7 \cdot 10^{x}+a$, where $x$ and $a$ are natural numbers with $a<10^{x}$. Due to the required property, we have

$$
k=7 \cdot 10^{x}+a=3(10 a+7)
$$

From this, we obtain by equivalent transformation

$$
a=\frac{7 \cdot 10^{x}}{29}-\frac{21}{29}
$$

Now we divide $7 \cdot 10^{x}$ by 29; we determine the number $x$ by continuing the division until the remainder 21 appears. This ensures that $a$ becomes a natural number. It results in

$$
a=241379310344827586206896551 \quad \text { and } \quad k=7241379310344827586206896551
$$

The verification confirms the correctness.