Properties of Family of 333,667 and Related Numbers
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Theorem
This page reports on certain properties, difficult to classify, of the number $333 \, 667$, and patterns arising.
Product with Certain Repetitive Numbers
| \(\ds 333 \, 667 \times 296\) | \(=\) | \(\ds 98 \, 765 \, 432\) | ||||||||||||
| \(\ds 33 \, 336 \, 667 \times 2996\) | \(=\) | \(\ds 99 \, 876 \, 654 \, 332\) | ||||||||||||
| \(\ds 3 \, 333 \, 366 \, 667 \times 29 \, 996\) | \(=\) | \(\ds 99 \, 987 \, 666 \, 543 \, 332\) |
| \(\ds 333 \, 667 \times 1113\) | \(=\) | \(\ds 371 \, 371 \, 371\) | ||||||||||||
| \(\ds 33 \, 336 \, 667 \times 11 \, 133\) | \(=\) | \(\ds 371 \, 137 \, 113 \, 711\) | ||||||||||||
| \(\ds 3 \, 333 \, 366 \, 667 \times 111 \, 333\) | \(=\) | \(\ds 371 \, 113 \, 711 \, 137 \, 111\) |
| \(\ds 333 \, 667 \times 2223\) | \(=\) | \(\ds 741 \, 741 \, 741\) | ||||||||||||
| \(\ds 33 \, 336 \, 667 \times 22 \, 233\) | \(=\) | \(\ds 741 \, 174 \, 117 \, 411\) | ||||||||||||
| \(\ds 3 \, 333 \, 366 \, 667 \times 222 \, 333\) | \(=\) | \(\ds 741 \, 117 \, 411 \, 174 \, 111\) |
Squares
The square of any number consisting of:
- a string of $3$s
followed by:
- a string of $6$s
followed by:
- a single $7$
has its digits all in an increasing sequence:
| \(\ds 333 \, 667^2\) | \(=\) | \(\ds 111 \, 333 \, 666 \, 889\) | ||||||||||||
| \(\ds 33 \, 366 \, 667^2\) | \(=\) | \(\ds 1 \, 113 \, 334 \, 466 \, 688 \, 889\) |
The same is true of numbers of the following form:
| \(\ds 16 \, 667^2\) | \(=\) | \(\ds 277 \, 788 \, 889\) | ||||||||||||
| \(\ds 333 \, 334^2\) | \(=\) | \(\ds 111 \, 111 \, 555 \, 556\) | ||||||||||||
| \(\ds 333 \, 335^2\) | \(=\) | \(\ds 111 \, 112 \, 222 \, 225\) |