Let's find the smallest and largest 8-digit multiple of 360, where each digit is unique.
It is evident that the sought smallest number $n=10 n^{\prime}$, and the largest one $N=10 N^{\prime}$ can be written in the form where $n^{\prime}, N^{\prime}$ are the smallest and largest 7-digit multiples of 36, respectively, with all distinct digits; in which the digit 0 does not appear.

Since $n^{\prime}, N^{\prime}$ are divisible by 36, they are also divisible by 9 and 4, and conversely, if they are divisible by both, then they are also divisible by their product, since 9 and 4 are relatively prime. As is known, divisibility by 4 depends only on the selection of the last two digits, including their order; in divisibility by 9, however, every digit of the number plays a role, but the order of the digits is not significant, the sum of the digits must be divisible by 9.

The sum of the digits in question is $1+2+\ldots+9=45$, which is divisible by 9. Two of these digits do not appear, and their sum must also be divisible by 9 and at most $9+8=17$, so the sum of the unused digits is 9. The possibilities for omission are:

1 and $8, \quad 2$ and $7, \quad 3$ and $6 \quad$ or 4 and 5.

We want to start $N^{\prime}$ with the largest possible digits, so we try to leave out the 4, 5 digits, which are the smallest of the possible larger ones. Let's try the $N^{\prime}$ starting with the largest remaining digits in the form $9876 \ldots$. If 2 of the remaining $1,2,3$ digits stand at the end of $N^{\prime}$, 1 and 3 in any order ensure divisibility by 4, so the largest of the question numbers is $N^{\prime}=9876312$ and $N=98763120$.

Similarly, in the case of $n^{\prime}$, we try to leave out the smaller of the possible digit pairs, again the 4, 5, and choose 123 as the first three digits of $n^{\prime}$. If the $6,7,8,9$ digits to be used end with 6, the tens digit in front of it must be odd, so the 7896 ending is the smallest, if it ends with 8, then an even tens digit must precede it, 7968 results, which is larger than the former. According to these, $n=12378960$.

Rákos Magdolna (Székesfehérvár, Teleki B. g. I. o. t.)