\section*{Exercise 1 - 281021}

The smallest positive natural number is sought, whose digit representation (in the decimal system) consists only of the digits 0 and 1 and which is divisible by 450.
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For natural numbers whose digit representation consists only of the digits 0 and 1, the following applies:

1. Such a number is positive if and only if the number of digits 1 is not zero.

2. It is divisible by 450 if and only if it is divisible by 9 and by 50, since 9 and 50 are coprime.

3. It is divisible by 50 if and only if its digit sequence ends in ...00; since divisibility by 50 is also possible with an ending ...50, but here only 0 and 1 occur.

4. It is divisible by 9 if and only if its cross sum is divisible by 9; its cross sum is here equal to the number of digits 1 in the digit representation.

The digit representation of the smallest number that satisfies all these conditions thus consists of exactly nine digits 1 followed by two digits 0. That is, the number sought is 11111111100.