## Task 1 - 150911

Determine all four-digit natural numbers written in the decimal system that simultaneously meet the following conditions!

(1) The number is written with four digits, each of which, read individually, represents four consecutive numbers. No requirements are placed on the order of these digits.

(2) The number is divisible by 99.
I. If a number satisfies the given conditions, then:

Due to (1), the number can be written with the digits $0,1,2,3$ or $1,2,3,4$ or $2,3,4,5$ or $3,4,5,6$ or $4,5,6,7$ or $5,6,7,8$, or $6,7,8,9$ (in any order).

Due to (2), the number must be divisible by 9. According to the divisibility rule for 9, the sum of its digits must be divisible by 9.

Since $0+1+2+3=5 ; 1+2+3+4=10 ; 2+3+4+5=14 ; 3+4+5+6=18 ; 4+5+6+7=22$; $5+6+7+8=26 ; 6+7+8+9=30$, this is only true if the number contains the digits 3, 4, 5, 6 (in any order).

There are exactly 24 such numbers, namely:

$3456,3465,3546,3564,3645,3654,4356,4365,4536,4563,4635,4653,5346,5364,5436,5463,5634$, $5643,6345,6354,6435,6453,6534,6543$.

Of these, only the 8 numbers

$3465,3564,4356,4653,5346,5643,6435,6534$, are divisible by 11.

II. These 8 numbers satisfy condition (1).

They are also divisible by 9 and 11, and since 9 and 11 are coprime, they are divisible by 99, thus satisfying condition (2).

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0977.jpg?height=194&width=951&top_left_y=1679&top_left_x=264)