Multiplicative Persistence/Examples/39

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Examples of Multiplicative Persistence

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


Proof

We have:

\(\text {(1)}: \quad\) \(\ds 3 \times 9\) \(=\) \(\ds 27\)
\(\text {(2)}: \quad\) \(\ds 2 \times 7\) \(=\) \(\ds 14\)
\(\text {(3)}: \quad\) \(\ds 1 \times 4\) \(=\) \(\ds 4\)


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

\(\ds 3 \times 8\) \(=\) \(\ds 24\)
\(\ds 2 \times 9\) \(=\) \(\ds 18\)


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

The result follows.

$\blacksquare$