Multiplicative Persistence/Examples/277,777,788,888,899

Examples of Multiplicative Persistence

$277 \, 777 \, 788 \, 888 \, 899$ is the smallest positive integer which has a multiplicative persistence of $11$.

Proof

We have:

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

\(\ds 2 \times 7^6 \times 8^6 \times 9^2\)

\(=\)

\(\ds 2 \times 117 \, 649 \times 262 \, 144 \times 81\)

\(\ds \)

\(=\)

\(\ds 4 \, 996 \, 238 \, 671 \, 872\)

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

\(\ds 4 \times 9^2 \times 6^2 \times 2^2 \times 3 \times 8^2 \times 7^2 \times 1\)

\(=\)

\(\ds 438 \, 939 \, 648\)

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

\(\ds 4^2 \times 3^2 \times 8^2 \times 9^2 \times 6\)

\(=\)

\(\ds 4 \, 478 \, 976\)

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

\(\ds 4^2 \times 7^2 \times 8 \times 9 \times 6\)

\(=\)

\(\ds 338 \, 688\)

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

\(\ds 3^2 \times 8^3 \times 6\)

\(=\)

\(\ds 27 \, 648\)

\(\text {(6)}: \quad\)

\(\ds 2 \times 7 \times 6 \times 4 \times 8\)

\(=\)

\(\ds 2688\)

\(\text {(7)}: \quad\)

\(\ds 2 \times 6 \times 8^2\)

\(=\)

\(\ds 768\)

\(\text {(8)}: \quad\)

\(\ds 7 \times 6 \times 8\)

\(=\)

\(\ds 336\)

\(\text {(9)}: \quad\)

\(\ds 3 \times 3 \times 6\)

\(=\)

\(\ds 54\)

\(\text {(10)}: \quad\)

\(\ds 5 \times 4\)

\(=\)

\(\ds 20\)

\(\text {(11)}: \quad\)

\(\ds 2 \times 0\)

\(=\)

\(\ds 0\)

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

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $277,777,788,888,899$