2021 JMPSC Sprint Problems/Problem 17

Revision as of 11:20, 11 July 2021 by Mathdreams (talk | contribs) (Solution)

Problem

What is the smallest positive multiple of $1003$ that has no zeros in its decimal representation?

Solution

Notice that $1002 \cdot n = 1000n + 2n$. Since $1000n$ always has $3$ zeros after it, we have to make sure $2n$ has $3$ nonzero digits, so that the last 3 digits of the number $1002n$ doesn't contain a $0$. We also need to make sure that $n$ has no zeros in its own decimal representation, so that $1000n$ doesn't have any zeros other than the last $3$ digits. The smallest number $n$ that satisfies the above is $56$, so the answer is $1002 \cdot 56 = \boxed{56112}$.

~Mathdreams

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