31. a) Find all integers that start with the digit 6 and decrease by 25 times when this digit is erased.

b) Prove that there do not exist integers that decrease by 35 times when the first digit is erased.
31. a) Let the desired number have $k+1$ digits; in this case, it has the form $6 \cdot 10^{k} + y$, where $y$ is a $k$-digit number (which may start with one or several zeros). According to the problem, we have:

$$
6 \cdot 10^{k} + y = 25 \cdot y
$$

from which it follows that

$$
y = \frac{6 \cdot 10^{k}}{24}
$$

This implies that $k$ cannot be less than 2 (otherwise, $6 \cdot 10^{k}$ would not be divisible by 24). For $k \geqslant 2$, the number $y$ turns out to be $25 \cdot 10^{k-2}$, i.e., it has the form $250 \ldots 0$. Therefore, all the desired numbers have the form

$$
(k-2) \text{ zeros }
$$

$6250 \ldots 0 (n=0,1,2, \ldots)$.

n MY:Ieti

b) Solving the problem: find a number starting with a given digit $a$ and decreasing by 35 times when this digit is erased, similarly to problem a), we arrive at the following equality:

$$
y = \frac{a \cdot 10^{k}}{34}
$$

where $y$ is an integer (see the solution to problem a)). However, this equality clearly cannot hold for any integers $a \leqslant 9$ and $k$.

Note. Analogously to the solutions of problems 31a) and b), it can be shown that a number starting with a known digit $a$ can decrease by an integer $b$ times when this digit is erased only if $b-1$ is a larger $\alpha$ number such that all prime factors of $b-1$, except for 2 and 5, also appear (and in no lesser degree) in the composition of the number $a$ (i.e., that $\frac{a}{b-1}$ is a proper fraction that can be converted into a finite decimal fraction). For example, no number can decrease by 85 times when the first digit is erased $(85-1=84$ is divisible by $3 \cdot 7$, and no digit can simultaneously be divisible by 3 and 7), while a number that decreases by 15 times when the first digit is erased must start with the digit 7 ( $15-1=14$ is divisible by 7). The general form of numbers that decrease by a given number $b$ times when the known first digit $a$ is erased can be easily found.