Persistent Numbers

We define a number's persistence as the number of steps required to reduce it to a single digit by multiplying all its digits together repeatedly. For example, 77 has a persistence of four because it requires four steps to reduce it to one digit:

\(77 \rightarrow 49 \rightarrow 36 \rightarrow 18 \rightarrow 8\)

  • The smallest number with a persistence of one is 10.
  • The smallest number with a persistence of two is 25.
  • The smallest number with a persistence of three is 39...
  • And the smallest number with a persistence of four is 77.

What is the smallest number of persistence nine?


Inspired by Martin Gardner.
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