Multiplicative Persistence/Examples/679

Examples of Multiplicative Persistence

$679$ is the smallest positive integer which has a multiplicative persistence of $5$.

Proof

We have:

\(\text {(1)}: \quad\)

\(\ds 6 \times 7 \times 9\)

\(=\)

\(\ds 378\)

\(\text {(2)}: \quad\)

\(\ds 3 \times 7 \times 8\)

\(=\)

\(\ds 168\)

\(\text {(3)}: \quad\)

\(\ds 1 \times 6 \times 8\)

\(=\)

\(\ds 48\)

\(\text {(4)}: \quad\)

\(\ds 4 \times 8\)

\(=\)

\(\ds 32\)

\(\text {(5)}: \quad\)

\(\ds 3 \times 2\)

\(=\)

\(\ds 6\)

The fact that it is the smallest can be demonstrated by brute-force methods.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $679$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $679$