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

(Created page with "==Problem== What is the smallest positive multiple of <math>1003</math> that has no zeros in its decimal representation? ==Solution== asdf")
 
(Solution)
Line 3: Line 3:
  
 
==Solution==
 
==Solution==
asdf
+
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>.
 +
 
 +
~Mathdreams

Revision as of 11:20, 11 July 2021

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

Invalid username
Login to AoPS