Number theory · Digits · Concatenation · Divisibility by 8 · Eventual periodicity
For a positive integer \(n\), let \(x_n\) be the number obtained by writing the decimal representations of \(1, 2, \dots, n\) one after another; for example \(x_{14} = 1234567891011121314\). Define \(f \colon \mathbb{N} \to \mathbb{N}_0\) by: \(f(n)\) is the smallest number of digits that must be erased from the decimal representation of \(x_n\) so that the digits which remain, read in their original order, form a number divisible by \(8\). (Erasing all the digits is allowed; the empty result counts as the number \(0\).)
Do there exist positive integers \(t\) and \(n_0\) such that \(f(n+t) = f(n)\) for every \(n \geq n_0\)?
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Serbian National Competition (Drzavno takmicenje) 2018, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source