3
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Arrange the digits from 0 to 9 into a number which is divisible by every integer from 12 to 21 inclusive.

(There are 2 right answers.)

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3
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The numbers are

2137458960 and 6179584320

I found these results by calculating the least common multiple $\operatorname{lcm}(12,\dotsc 21) = 21162960$, then checking all multiples of this value up to 9876543210 with a script: https://jsfiddle.net/qdnyLfx8/

There must be more elegant solutions that work without using a computer.

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    $\begingroup$ Well then I won't post the 2nd number. $\endgroup$ – Joel Rondeau Aug 16 '16 at 4:11
  • $\begingroup$ The prime factors of 12 to 21 are 24×3×5×7×13×17×19, therefore the two 10-digit pandigital numbers are a multiple of 7054320. 2137458960 [7054320×303] and 6179584320 [7054320×876]. $\endgroup$ – Jamal Senjaya Aug 16 '16 at 4:29
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    $\begingroup$ @JamalSenjaya Based on GOTO 0's LCM calculation you seem to have missed a factor of 3 somewhere. $\endgroup$ – user253751 Aug 16 '16 at 8:46
2
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If I take all the factors of 12-21 and multiply them together, I get

5 * 7 * 9 * 13 * 16 * 17 * 19, or
21162960

So now all that is left to do is to keep adding it to itself until I get a 10 digit number that satisfies the requirements. The first such number is

2137458960

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