# 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 – GOTO 0 Dec 30 '14 at 3:48
• Wow. All hail the greatest Wiki. – James Lynch Dec 30 '14 at 3:58
• Fun fact; Any multiple of 9, when reducing to a single digit with this method, will return 9! – warspyking Dec 30 '14 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. – Chris Dec 30 '14 at 12:08
• I think he means the en.wikipedia.org/wiki/Digital_root – Ivo Beckers Dec 30 '14 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