Multiplicative Persistence/Examples/39

Examples of Multiplicative Persistence
$39$ is the smallest positive integer which has a multiplicative persistence of $3$.

Proof
We have:

The product of the $2$ digits of a positive integer smaller than $39$ is less than $25$, for example:

From Multiplicative Persistence of 25, all such positive integers have a multiplicative persistence of $2$.

The result follows.