# Martin Gardner - Persistence

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

The smaller of persistence four is 77

What is the smallest number of persistence five?

• Wikipedia knows everything: Persistence of a number Dec 30, 2014 at 3:48
• Wow. All hail the greatest Wiki. Dec 30, 2014 at 3:58
• Fun fact; Any multiple of 9, when reducing to a single digit with this method, will return 9! Dec 30, 2014 at 4:20
• @warspyking: Am I missing something? 18 is a multiple of 9. When applying this I multiply all its digits to get 8. 8 is not equal to 9. Dec 30, 2014 at 12:08
• I think he means the en.wikipedia.org/wiki/Digital_root Dec 30, 2014 at 12:29

Brute forcing with Lua gave me

679

This is also confirmed by Wikipedia and OEIS:

http://en.m.wikipedia.org/wiki/Persistence_of_a_number

http://oeis.org/A003001

This is a list of the smallest numbers of persistence $n$ with ($1\leq n\leq11)$.

1 10

2 25

3 39

4 77

5 679 ---> (answer of this problem)

6 6788

7 68889

8 2677889

9 26888999

10 3778888999

11 277777788888899