Number theory · Decimal representation · Digits · Divisibility · Repunits
Find all digits \(n\) and all \(2018\)-digit positive integers \(x = \overline{a_{2017} \ldots a_2 a_1 a_0}\) for which
\[ n \cdot x = \overline{(a_{2017} + n) \ldots (a_2 + n)(a_1 + n)(a_0 + n)} . \]
Here the bar denotes decimal notation, so the right-hand side is the number obtained from \(x\) by increasing every one of its digits by \(n\); in particular each \(a_i + n\) is required to be a single digit.
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Serbian Regional Competition 2018, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source