Number theory · Factorials · Legendre formula · Prime valuations · Trailing zeros · Decimal representation
Does there exist a natural number \(n\) for which the decimal expansion of \(n!\) has the form
\[ n! = \ldots 2012\,\underbrace{00\ldots 0}_{k}, \]
that is, ends in the digit block \(2012\) followed by exactly \(k\) zeros, for some natural number \(k\)?
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Serbian Municipal Competition 2013, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source