Multiplicative Persistence/Examples/77
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Examples of Multiplicative Persistence
$77$ is the smallest positive integer which has a multiplicative persistence of $4$.
Proof
We have:
| \(\text {(1)}: \quad\) | \(\ds 7 \times 7\) | \(=\) | \(\ds 49\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds 4 \times 9\) | \(=\) | \(\ds 36\) | |||||||||||
| \(\text {(3)}: \quad\) | \(\ds 3 \times 6\) | \(=\) | \(\ds 18\) | |||||||||||
| \(\text {(4)}: \quad\) | \(\ds 1 \times 8\) | \(=\) | \(\ds 8\) |
$39$ is the smallest positive integer with multiplicative persistence of $3$.
Hence if the product of digits of $n$ is less than $39$, its multiplicative persistence cannot exceed $3$.
Therefore we only need to check:
- $58, 59, 67, 68, 69, 76$
and we have:
| \(\text {(1)}: \quad\) | \(\ds 5 \times 8\) | \(=\) | \(\ds 40\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds 4 \times 0\) | \(=\) | \(\ds 0\) | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds 5 \times 9\) | \(=\) | \(\ds 45\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds 4 \times 5\) | \(=\) | \(\ds 20\) | |||||||||||
| \(\text {(3)}: \quad\) | \(\ds 2 \times 0\) | \(=\) | \(\ds 0\) | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds 6 \times 7 = 7 \times 6\) | \(=\) | \(\ds 42\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds 4 \times 2\) | \(=\) | \(\ds 8\) | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds 6 \times 8\) | \(=\) | \(\ds 48\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds 4 \times 8\) | \(=\) | \(\ds 32\) | |||||||||||
| \(\text {(3)}: \quad\) | \(\ds 3 \times 2\) | \(=\) | \(\ds 6\) | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds 6 \times 9\) | \(=\) | \(\ds 54\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds 5 \times 4\) | \(=\) | \(\ds 20\) | |||||||||||
| \(\text {(3)}: \quad\) | \(\ds 2 \times 0\) | \(=\) | \(\ds 0\) |
so none of those numbers have a multiplicative persistence of $4$.
$\blacksquare$