A positive integer has 73 digits, all different from zero. Prove that we can erase 64 digits such that the new number is divisible by 37.
Rmc, 2020
Solution
After the elimination of the 64 digits the final number has digits.
If in the composition of , every digit appears at most 8 times, then would have at most digits, which is false.
Therefore, there is a digit different from 0, let that be , that appears 9 times in . Then we can erase 64 digits from such that the final number would have the form , so the number is divisible by 37.
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