## Task 1 - 250911

From the digits 1, 9, 8, 5, all possible four-digit numbers are to be formed, where each of the digits appears exactly once in each of these numbers.

Determine the number of such numbers that are

a) divisible by 2,

b) divisible by 3,

c) divisible by 4,

d) divisible by 5,

e) divisible by 6,

and list these numbers!
a) Among the four-digit numbers mentioned, exactly those ending in an even digit, i.e., 8, are divisible by 2. With the remaining 3 digits, 6 different numbers can be formed, so there are exactly 6 such numbers divisible by 2, namely 1598, 1958, 5198, 5918, 9158, and 9518.

b, e) Since the cross sum of 1985 is not divisible by 3, and this also applies to all numbers formed by rearrangement, none of the numbers are divisible by 3 and thus none are divisible by 7.

c) A number is divisible by 4 if and only if the number formed by the last two digits is divisible by 4. However, from the digits 1, 5, 8, and 9, no such two-digit number can be formed; it would have to be even, i.e., end in 8, and the numbers 18, 58, and 98 are not divisible by 4. Therefore, there are no such numbers divisible by 4.

d) Among these numbers, exactly those ending in 5 are divisible by 5. With the remaining 3 digits, 6 different numbers can be formed. Thus, exactly 6 such numbers are divisible by 5, namely 1895, 1985, 8195, 8915, 9185, and 9815.