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Difference between revisions of "2021 JMPSC Sprint Problems/Problem 17"

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==Solution==
 
==Solution==
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Notice that <math>1002 \cdot n = 1000n + 2n</math>. Since <math>1000n</math> always has <math>3</math> zeros after it, we have to make sure <math>2n</math> has <math>3</math> nonzero digits, so that the last 3 digits of the number <math>1002n</math> doesn't contain a <math>0</math>. We also need to make sure that <math>n</math> has no zeros in its own decimal representation, so that <math>1000n</math> doesn't have any zeros other than the last <math>3</math> digits. The smallest number <math>n</math> that satisfies the above is <math>56</math>, so the answer is <math>1002 \cdot 56 = \boxed{56112}</math>.
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Notice that <math>1003 \cdot n = 1000n + 3n</math>. Since <math>1000n</math> always has <math>3</math> zeros after it, we have to make sure <math>3n</math> has <math>3</math> nonzero digits, so that the last 3 digits of the number <math>1003n</math> doesn't contain a <math>0</math>. We also need to make sure that <math>n</math> has no zeros in its own decimal representation so that <math>1000n</math> doesn't have any zeros other than the last <math>3</math> digits. The smallest number <math>n</math> that satisfies the above is <math>\frac{111}{3}=37</math>, so the answer is <math>1003 \cdot 37 = \boxed{37111}</math>.
  
 
~Mathdreams
 
~Mathdreams
 +
~edited by tigerzhang

Revision as of 11:53, 11 July 2021

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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