Multiplicative Persistence/Examples/25

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.

$\blacksquare$