2. Can the number 2019 be represented as the sum of 90 natural numbers with the same digit sum?

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Solution. Each addend has the same sum of digits, so their remainders when divided by 9 are the same (the sum of the digits of a number gives the same remainder when divided by 9 as the number itself). The sum of 90 identical remainders is divisible by 9, which means the sum of any 90 natural numbers with the same sum of digits is divisible by 9. But the number 2019 is not divisible by 9. Answer: no.

Comment. Similar reasoning about remainders when divided by 3 (see problem №2 for 10th grade) does not lead to the correct solution, since 2019 is divisible by 3 but not by 9.