Properties of 5,559,060,566,555,523

Theorem

$3^{33} = 5 \, 559 \, 060 \, 566 \, 555 \, 523$ has the following properties:

It has a remarkably large number of $5$s (half of its digits).

Multiply it by $2$, $4$ or $6$, and the result has $10$ of a particular digit.

Multiply it by $8$, and $9$ of the digits of the result are $4$.

$\ds 5 \, 559 \, 060 \, 566 \, 555 \, 523 \times 2$

$=$

$\ds 11 \, 118 \, 121 \, 133 \, 111 \, 046$

$10$ $1$s

$\ds 5 \, 559 \, 060 \, 566 \, 555 \, 523 \times 4$

$=$

$\ds 22 \, 236 \, 242 \, 266 \, 222 \, 092$

$10$ $2$s

$\ds 5 \, 559 \, 060 \, 566 \, 555 \, 523 \times 6$

$=$

$\ds 33 \, 354 \, 363 \, 399 \, 333 \, 138$

$10$ $3$s

$\ds 5 \, 559 \, 060 \, 566 \, 555 \, 523 \times 8$

$=$

$\ds 44 \, 472 \, 484 \, 532 \, 444 \, 184$

$9$ $4$s

$\blacksquare$

David Wells reports in his Curious and Interesting Numbers, 2nd ed. of $1997$ that this result is attributed to David Roberts, but no details are given.