[ Decimal numeral system ] [Examples and counterexamples. Constructions

## Authors: Khenoarov R.G, Bogdanov I.I.

There are 19 cards. Is it possible to write a non-zero digit on each card so that from these cards exactly one 19-digit number divisible by 11 can be formed?

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We will write the digit 2 on ten cards, and the digit 1 on the remaining nine cards. It is known that a natural number is divisible by 11 if and only if the alternating sum \( S \), formed from the digits of the number, is a multiple of 11.

In a number composed of ten digits 2 and nine digits 1, the inequalities \(-7 \leq S \leq 11\) are satisfied.

The sum of all digits is odd (it is equal to 21), so \( S \) is also odd. From -7 to 11, there is only one odd number that is a multiple of 11, which is the number 11. But for

\( S=11 \) there is only one possibility - when twos are in the odd positions and ones are in the even positions.

## Answer

It is possible.