Multiplicative Persistence/Examples/25
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Examples of Multiplicative Persistence
$25$ is the smallest positive integer which has a multiplicative persistence of $2$.
Proof
Trivially:
\(\text {(1)}: \quad\) | \(\ds 2 \times 5\) | \(=\) | \(\ds 10\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds 1 \times 0\) | \(=\) | \(\ds 0\) |
All positive integers between $10$ and $19$ are seen to have a multiplicative persistence of $1$:
- $1 \times n = n$
where $n$ is a single digit.
Then for 2-digit positive integers starting with $2$:
- $2 \times n > 9 \implies n > 4$
by inspection.
Hence the result.
$\blacksquare$