Difference between revisions of "2021 JMPSC Sprint Problems/Problem 17"
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==Solution== | ==Solution== | ||
| − | Notice that <math> | + | 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 | ||
| + | |||
| + | ==See also== | ||
| + | #[[2021 JMPSC Sprint Problems|Other 2021 JMPSC Sprint Problems]] | ||
| + | #[[2021 JMPSC Sprint Answer Key|2021 JMPSC Sprint Answer Key]] | ||
| + | #[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]] | ||
| + | {{JMPSC Notice}} | ||
Latest revision as of 16:15, 11 July 2021
Problem
What is the smallest positive multiple of 
that has no zeros in its decimal representation?
Solution
Notice that 
. Since 
always has 
zeros after it, we have to make sure 
has nonzero digits, so that the last 3 digits of the number 
doesn't contain a 
. We also need to make sure that 
has no zeros in its own decimal representation so that doesn't have any zeros other than the last digits. The smallest number that satisfies the above is 
, so the answer is 
.
~Mathdreams
~edited by tigerzhang
See also
The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition. 
