2021 JMPSC Sprint Problems/Problem 17

Problem

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

Solution

Notice that $1003 \cdot n = 1000n + 3n$ . Since $1000n$ always has $3$ zeros after it, we have to make sure $3n$ has $3$ nonzero digits, so that the last 3 digits of the number $1003n$ 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 $\frac{111}{3}=37$ , so the answer is $1003 \cdot 37 = \boxed{37111}$ .

~Mathdreams

~edited by tigerzhang

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2021 JMPSC Sprint Problems/Problem 17

Problem

Solution