1. What is the smallest number of digits that can be erased from the number 20162016 so that the result is divisible by 2016 (it is not allowed to erase nothing)? Please note that you need to not only provide an example but also explain why it is impossible to do with fewer digits.
Answer. Three. Solution. Since 2016 is divisible by 9, the sum of the digits of the resulting number after erasing some digits must also be divisible by 9. The sum of the digits of the number 20162016 is 18. Erasing one or two zeros does not yield the desired result: the numbers 2162016, 2016216, and 216216 are not divisible by 2016. Therefore, we need to erase digits that sum up to 9. Since the sum of any two digits of the number 20162016 is less than 9, we will have to erase at least three digits. Three digits can be erased: $2016 z 016=20160$.