## 104. Math Puzzle $1 / 74$

Determine the smallest natural number $z$ that ends in 4. If you remove the 4 at the end and place it at the front, you get four times $z$.
Let $z=x \ldots y 4$ be the smallest natural number that ends in 4. According to the problem statement, $4 z=4 x \ldots y=(x \ldots y 4) \cdot 4$.

By continuous multiplication, one obtains $z$ by inserting the digits obtained from the partial products into the second factor. The multiplication starts with $4 \cdot 4=16$. The 6 is then written instead of $y$. The next partial multiplication is then $4 \cdot 6=24$ plus 1 from 16, resulting in 25. The 5 is then written in the product and the factor, just like the 6. The multiplication stops when a 4 is obtained in the product without a carry. $z=102564$ is the desired number.