(Answered) There are 1000 natural numbers a1 a2 a1000 prove that of

Statistics

There are 1000 natural numbers a₁, a₂,...., a1000. Prove that of the numbers 3^a₁, 3^a₂,...., 3^{a₁₀₀₀}, there must be 56 of them which have the same remainder when divided by 19.
(Hint: this is not a misprint. If the best you can do is prove the statement with 56 replaced by 53, then you're missing something.)

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