4. Determine the largest two-digit number $k$ with the following property: there exists a natural number $N$, from which, after erasing the first digit from the left, we get a number $k$ times smaller. (After erasing the digit, the number representation may start with one or several zeros.) For the determined number $k$, then find the smallest suitable number $N$.

The regional round in category C takes place

## on Tuesday, March 27, 2007

so that it starts in the morning and contestants have 4 hours of clean time to solve the problems. For each problem, a contestant can earn 6 points, and a successful solver is a student who scores 10 points or more. Allowed aids are writing and drawing supplies, school MF tables, and calculators without a graphical display. These details will be communicated to the students before the start of the competition.
4. Any $(m+1)$-digit natural number $N$ with the first digit $c$ has the expression $N = c \cdot 10^{m} + x$, where $x$ is precisely the number obtained from $N$ by removing the first digit $c$. According to the problem, it must hold that $N = c \cdot 10^{m} + x = kx$ or $c \cdot 10^{m} = (k-1)x$. The number $k-1$ must therefore be a divisor of the number $c \cdot 10^{m}$, which has only single-digit prime factors: the prime numbers 2, 5, and the prime factors of the digit $c$. We will therefore test the prime factors of the number $k-1$ for the largest two-digit $k$:

$\triangleright k=99: k-1=98=2 \cdot 7^{2}$ does not satisfy, because $7^{2} \nmid c \cdot 10^{m}$.

$\triangleright k=98: k-1=97$ does not satisfy, because 97 is a two-digit prime number.

$\triangleright k=97: k-1=96=2^{5} \cdot 3$ satisfies, because, for example, $2^{5} \cdot 3 \mid c \cdot 10^{m}$ for $c=3$ and $m=5$; to get a smaller $N$, we can choose a smaller $m=4$ and $c=3 \cdot 2=6$ (other $c$ for $m=4$ do not satisfy). For $m \leq 3$, the relation $2^{5} \cdot 3 \mid c \cdot 10^{m}$ does not hold for any non-zero digit $c$.

The largest two-digit $k$ we are looking for is therefore 97. According to the previous discussion, we determine the smallest suitable $N$, which corresponds to $m=4, c=6$, and $x=6 \cdot 10^{4} : 96 = 625$, so $N = 6 \cdot 10^{4} + 625 = 60625$.

Answer: The sought $k$ is 97 and the smallest suitable $N$ is 60625.

For a complete solution, award 6 points, of which 3 points for the general finding that the number $k-1$ must be a divisor of the product of a non-zero digit and a power of 10. A complete solution is also considered if the values $k=99$, $k=98$ are excluded by separate procedures and for the value $k=97$ the smallest suitable $N$ is determined (not guessed). In a complete solution, if a suitable but not the smallest possible $N$ (for example, $N=303125$) is given for $k=97$, award 5 points.