Let's find such numbers that when multiplied by 12345679, we get numbers consisting of 9 identical digits.
A 9-digit number consisting of equal digits has the general form 111111111a, where $a$ can take any value from 1 to 9. Therefore,

$$
12345679 x = 111111111 a
$$

from which

$$
x = 9 a
$$

the sought numbers are thus:

$$
9, 18, 27, 36, 45, 54, 63, 72, 81
$$

(Raab Rezsó, Györ.)

Number of solutions: 22.