Number theory · Digit sums · Divisibility rules · Counterexamples
Decide whether the following claim is true, and prove your answer.
For every positive integer \(n\) there exists a positive integer \(x\) such that all three of the following hold: \(x\) is divisible by \(n\); the digits of \(x\) add up to \(n\); and the decimal representation of \(x\) ends with the block of digits that forms the decimal representation of \(n\).
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Serbian Regional Competition 2010, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source