Prove that there exist two distinct natural numbers m and n with m, n ≤ 2049 such that 19m − 19n is divisible by 2019

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Prove that there exist two distinct natural numbers m and n with m, n ≤ 2049 such that 19m − 19n is divisible by 2019. The truth of this claim does not depend on the exact values 19, 2019, and 2049. So, your solution should give an argument that is easy to adapt to other values for these constants rather than, say, using a computer program to find specific values for m and n.

I assume this question requires the pigeon hole principle as well as modular arithmetic.