Product with Repdigit can be Split into Parts which Add to Repdigit/Mistake

Source Work

1986: David Wells: Curious and Interesting Numbers:

The Dictionary

$6666$

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary

$6666$

Mistake

... if a number is multiplied by a number whose digits are all the same, for example, let $894$ be multiplied by $22,222$, then in this case the right-hand $5$ digits, added to the left-hand portion, form another number with equal digits: $894 \times 22,222 = 19866468$ and $198 + 66,468 = 66666$.

This is only guaranteed to work when the repdigit is strictly longer than the non-repdigit part.

For example:

\(\ds 19 \, 485 \times 222\)

\(=\)

\(\ds 4 \, 325 \, 670\)

\(\ds 4325 + 670\)

\(=\)

\(\ds 4995\)

It sometimes works, but you can't guarantee it:

\(\ds 19 \, 485 \times 2222\)

\(=\)

\(\ds 43 \, 295 \, 670\)

\(\ds 4329 + 5670\)

\(=\)

\(\ds 9999\)

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $6666$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6666$