digit 0 ( # 16 )
1010600000 = 10432 * 96875
6012000000 = 64128 * 93750
digit 1 ( # 11 )
1111011128 = 12403 * 89576
6111141113 = 64273 * 95081
digit 2 ( # 24 )
2262126222 = 23046 * 98157
5222222258 = 56738 * 92041
digit 3 ( # 17 )
3334733336 = 34718 * 96052
3931533333 = 54093 * 72681
digit 4 ( # 21 )
1148444444 = 13657 * 84092
5444424440 = 69524 * 78310
digit 5 ( # 10 )
1555585551 = 26403 * 58917
7555551552 = 83472 * 90516
digit 6 ( # 7 )
4666636664 = 54017 * 86392
6766616636 = 79201 * 85436
digit 7 ( # 2 )
complete set !
2777379777 = 32091 * 86547
7797777872 = 81736 * 95402
digit 8 ( # 5 )
1888388388 = 20157 * 93684
7688288888 = 80612 * 95374
digit 9 ( # 0 )
nihil
A _twofold_ solution pops up if we keep the seven identical
digits in one uninterrupted cluster.
Note that it is a very beautiful construction since the two
5-digit factors are 'digital anagrams' amongst themselves.
5222222258 = 56738 * 92041 7055555558 = 76358 * 92401 |
56738 * 92041
x x
76358 * 92401
Two 10-digits contain more than seven identical digits
i.e. eight identicals occurring with digit 6 in both cases
(no higher sequences exists). Quite a Beastly affair !
Concatenate the remaining digits and you'll agree with me
that 2 0 0 5 was the best year to discover this in.
My destiny with the World!Of Numbers is on schedule !
6626666660 = 72308 * 91645 6666606665 = 79021 * 84365 |
Restricting the remaining digits to be identical leads us
finally to the next three nice solutions.
2222323232 = 29104 * 76358 5553355355 = 67391 * 82405 6000660000 = 63750 * 94128 |
The third solution again reveals to us the presence of
666 or the Number of the Beast !
The extended versions of the Beast are available as well !
For instance as differences between the two 5-digit factors.
4690873152 = 65238 * 71904 and 71904 65238 = 6666 |
1802967435 = 20649 * 87315 and 87315 20649 = 66666 |
To conclude this expansion on the subject two equations
yielding palindromes, the first one is of a repdigit kind,
our 5-digit Beast turned topsyturvy !
2495671308 = 47931 * 52068 and 47931 + 52068 = 99999 |
2846031795 = 30645 * 92871 and 92871 30645 = 62226 |
The palindrome 62226 will come back in another format,
so keep it in mind...
**
Inspired by B.S. Rangaswamy's book I set out to look for
ninedigital numbers that are the product of two 5-digit factors
and that taken together form pandigitals.
141 solutions came up which is a palindromic total !
The smallest one is
315867942 = 15486 * 20397
The largest one is
987561234 = 28179 * 35046
The uniqueness of the following result is that
the addition of the two factors is palindromic.
And we came across that one before, didn't we...
857264193 = 20589 * 41637 and 20589 + 41637 = 62226 |
The Beast took refuge in one of the 141 solutions !
The result is composed of the Number of the Beast and
the digitsum of 666 i.e., 6 + 6 + 6 = 18 !
846173952 = 17082 * 49536 and 17082 + 49536 = 66618 |
**
Episode 55 : Six Soldiers (p. 77)
This chapter prompts you explore 10 digit numbers having six
identical numerals positioned in a continuous line.
While recomputing all the possible solutions (total of 44),
including those with two 5-digit factors not evenly
divisible by 3, I stumbled over the following curios ¬
Particularly beautiful is this item because it
uses only two distinct digits namely '3' & '0'.
3333330000 = 34125 * 97680
Some not continuous solutions with only two distinct digits
are the following three items (from a total of 1581 - 44 or 1537).
1111551155 = 23705 * 46891
4442244242 = 45317 * 98026
5566656665 = 62473 * 89105
Also special is this item because the digits of the
tendigit number are in 'ascending order' and consecutive !
6777777888 = 72561 * 93408
**
Awesome x anagrammatical x equations
emerged while comparing various factors from
nine- and pandigital output lists
The first construct is with multiplicand 27489 and his
anagram mate 49278. The same couple of multipliers
can be applied to arrive at nine- and pandigital numbers.
Note that 27489 + 49278 = 76767 and palindromic !
27489 * | ↗ ↘ | 5361(0) = 147368529(0) x x 6351(0) = 174582639(0) |
x |
49278 * | ↗ ↘ | 5361(0) = 264179358(0) x x 6351(0) = 312964578(0) |
A second construct produces a ninedigital and a pandigital with
this couple of multiplicand and multiplier anagrams.
ps. the second 49278 was also used in the above setup !
The third multiplication with the palindromic outcome
and the fourth equation with a heptadic result
finish this illustration of our four interrelated concepts
in a wonderful and astonishing manner [ Dec 4, 2005 ].
Finally three _still interesting_ leftovers from my search.
24561, 45618 & 61329 are the resp. multiplicands.
Note that in the third case the largest 4-digit multiplier
is the reversal of the smallest 4-digit multiplier.
24561 * | ↗ ↘ | 8739(0) = 214638579(0) x x 8793(0) = 215964873(0) |
45618 * | ↗ ↘ | 3792(0) = 172983456(0) x x 9372(0) = 427531896(0) |
61329 * | ↗ ↘ | 7458(0) = 457391682(0) x x 8547(0) = 524178963(0) |
**
Palindromes as products of two 5-digit factors.
There are only ten palindromes consisting of nine digits
and three palindromes consisting of ten digits that are
the products of two 5-digit factors which taken together
form a pandigital. Prime factors are highlighted.
all odd digits! 393555393
385454583
690555096
431292134
707595707
629979926
919222919
966737669
all even digits! 804464408
883000388
2936556392
4461771644
4878998784 |
= 10857 * 36249
= 13569 * 28407
= 15708 * 43962
= 16978 * 25403
= 17563 * 40289
= 19658 * 32047
= 25471 * 36089
= 25801 * 37469
= 25897 * 31064
= 28517 * 30964
= 32564 * 90178
= 51029 * 87436
= 53724 * 90816 |
One anagrammatical combination shows up here.
10857 * 36249 = 393555393
x x
15708 * 43962 = 690555096
|
Palindromes as products of a 4-digit and a 5-digit factor.
There are three palindromes consisting of eight digits and
thirtythree palindromes consisting of nine digits that are
the products of a 4-digit and a 5-digit factors which taken
together form a ninedigital.
Five double anagrammatical combinations also show up here.
1453 * 29678 = 43122134
x x
4351 * 68792 = 299313992
|
9256 * 87143 = 806595608
x x
9526 * 87413 = 832696238
|
4197 * 28563 = 119878911
x x
9417 * 62835 = 591717195
|
4659 * 82137 = 382676283
x x
6945 * 78321 = 543939345
|
5824 * 79136 = 460888064
x x
8425 * 61793 = 520606025
|
One triple anagrammatical combination exists as well !
2964 * 71358 = 211505112
x x
4629 * 81537 = 377434773
x x
4926 * 53187 = 261999162
|
**
Mixed anagram equations starting from a common factor.
The first and the last two constructions are remarkable in the
sense that all their factors are also anagrams among each other !
Many examples I found are displayed as well but without
the pretention of having made an 'exhaustive' list.
**
Palindromic lookalikes.
Behold the next two palindromes and get enchanted.
The first equation is an 8-digit palindrome expressed
as a product of a 4- and a 5-digit factor which taken
together form a ninedigital.
The second equation is a 9-digital palindromic anagram
from the previous one except for the extra zero digit 0
expressed as a product of two 5-digit factors which
taken together form a pandigital.
A zero is also what is needed to make the crossing
from ninedigital to pandigital numbers !
8803_3088 = 4576 * 19238
8830_0_0388 = 28517 * 30964
Compare the next two palindromes and get enchanted.
The first equation is an 8-digit palindrome expressed
as a product of a 4- and a 5-digit factor which taken
together form a ninedigital.
The second equation is a 9-digit palindrome identical
to the previous one except for the extra middle digit 9
expressed as a product of two 5-digit factors which
taken together form a pandigital.
4312_2134 = 1453 * 29678
4312_9_2134 = 16978 * 25403
There exist imho no better constructions that can
synthesize this wonplate in such a beautiful way !
**
Scintillating equations with two 5-digit factors and their reversals.
Below is shown an exceptionally nice pair of equations.
The two 5-digit factors are each other's reversals.
A truly _unique_ pandigital phenomenon !
4905361782 =
52137 * 94086
↔↕ ↕↔
73125 * 68049 =
4976083125 |
**