# 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?

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