Number theory · Digit sums · Prime powers · Modular arithmetic
For a natural number \(k\), let \(S(k)\) denote the sum of its digits. Do there exist natural numbers \(n\) and \(m\) such that
\[ S(n) \cdot S(n+1) \cdot \ldots \cdot S(n+m) = 2011^{2010}\,? \]
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Serbian Municipal Competition 2011, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source