First I have to explain what is meant by "multiplicative persistence". Don't worry: there is a simple concept behind this difficult name.
Multiplicative persistence
If you take a random number and multiply its digits, you always get a smaller number than the original one. If you repeat this a few times, then you end up with a one-digit number quite quickly. How many times you need to get to that one digit is called the multiplicative persistence.
For example: take the number 1799.
We multiply 1 × 7 × 9 × 9 = 567
Again, now with 567: 5 × 6 × 7 = 210
And again: 2 × 1 × 0 = 0
The persistence in this case is 3.
If you are going to try this out with numbers of your own choice, then you will discover that - even with large numbers - you usually end up with a fairly low multiplicative persistence.
The number 277777788888899
It appears that this number is the smallest number that has a multiplicative persistence of 11.
Is that so special?
Yes, because no number is known with a multiplicative persistence higher than 11.
Number systems
Of course, the multiplicative persistence also depends on which number system you work in. After all, in the binary system every number above 1 has a persistence of 1. In the ternary system almost every number has a 0 or a 1. Because of this that system has a maximum persistence of 3. The smallest number with that highest persistence ( which is 3 ) is: 2223. Check: 2223 → 223 → 113 → 1