Consecutive Triple of Repeated Digit-Products
Jump to navigation
Jump to search
Theorem
The triplet of integers $281, 282, 283$ have the property that if their digits are multiplied, and the process repeated on the result until only $1$ digit remains, that final digit is the same for all three, that is, $6$.
There does not exist an set of four consecutive integers which also all end up at the same single digit.
Proof
\(\ds 281: \ \ \) | \(\ds 2 \times 8 \times 1\) | \(=\) | \(\ds 16\) | |||||||||||
\(\ds 1 \times 6\) | \(=\) | \(\ds 6\) | ||||||||||||
\(\ds 282: \ \ \) | \(\ds 2 \times 8 \times 2\) | \(=\) | \(\ds 32\) | |||||||||||
\(\ds 3 \times 2\) | \(=\) | \(\ds 6\) | ||||||||||||
\(\ds 283: \ \ \) | \(\ds 2 \times 8 \times 3\) | \(=\) | \(\ds 48\) | |||||||||||
\(\ds 4 \times 8\) | \(=\) | \(\ds 32\) | ||||||||||||
\(\ds 3 \times 2\) | \(=\) | \(\ds 6\) |
$\blacksquare$
Historical Note
This result was attributed to Douglas J. Lanska and Charles Ashbacher, according to David Wells, who sourced the result from Journal of Recreational Mathematics, volume $22$ page $70$.
It has not so far proved possible to corroborate this.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $281$