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# Persistence

## Question

A number's persistence is the number of steps required to reduce it to a single digit by multiplying all its digits to obtain a second number, then multiplying all the digits of that number to obtain a third number, and so on until a one-digit number is obtained. For example, 77 has a persistence of four because it requires four steps to reduce it to one digit: 77-49-36-18-8. The smallest number of persistence one is 10, the smallest of persistence two is 25, the smallest of persistence three is 39, and the smaller of persistence four is 77. What is the smallest number of persistence five?

## 2 answers to this question

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Spoiler

679: 679 --> 378 --> 168 --> 48 --> 32 --> 6

Then for what it's worth, here are the minimums with persistence 6-9 (notice how they all build on each other):

• 6788: 6788 --> 2688 --> 768 --> 336 --> 54 --> 20 --> 0
• 68889: 68889 --> 27648 --> 2688 --> 768 --> 336 --> 54 --> 20 --> 0
• 2677889: 2677889 --> 338688 --> 27648 --> 2688 --> 768 --> 336 --> 54 --> 20 --> 0
• 26888999: 26888999 --> 4478976 --> 338688 --> 27648 --> 2688 --> 768 --> 336 --> 54 --> 20 --> 0

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Spoiler

The minimums for 10 and 11 are 3778888999 and 277777788888899

Edited by rocdocmac
grammar!

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