Difference between revisions of "2021 JMPSC Sprint Problems/Problem 17"
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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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| + | ~Mathdreams | ||
Revision as of 10:20, 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
