Multiplicative Persistence/Examples/77

Examples of Multiplicative Persistence
$77$ is the smallest positive integer which has a multiplicative persistence of $4$.

Proof
We have:

$39$ is the smallest positive integer with multiplicative persistence of $3$.

Hence if the product of digits of $n$ is less than $39$, its multiplicative persistence cannot exceed $3$.

Therefore we only need to check:
 * $58, 59, 67, 68, 69, 76$

and we have:

so none of those numbers have a multiplicative persistence of $4$.