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.
$\blacksquare$