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Saturday, July 26, 2008

Notable Number

There is a unique number of ten ciphers, for which the following holds:

* all ciphers from 0 up to 9 occur exactly once in the number;
* the first cipher is divisible by 1;
* the number formed by the first two ciphers is divisible by 2;
* the number formed by the first three ciphers is divisible by 3;
* the number formed by the first four ciphers is divisible by 4;
* the number formed by the first five ciphers is divisible by 5;
* the number formed by the first six ciphers is divisible by 6;
* the number formed by the first seven ciphers is divisible by 7;
* the number formed by the first eight ciphers is divisible by 8;
* the number formed by the first nine ciphers is divisible by 9;
* the number formed by the ten ciphers is divisible by 10.

First Question: Which number is this?

Second Question: There is a unique number of which the square and the cube together use all ciphers from 0 up to 9 exactly once. Which number is this?


First Answer: We construct the number cipher by cipher.

Tenth cipher
A number is divisible by 10 if it ends on a 0. Therefore, the tenth cipher of the requested number must be a 0.

Fifth cipher
A number is divisible by 5 if it ends on a 0 or 5. The 0 has already been used, so the fifth cipher of the requested number is a 5.

First cipher
A number is always divisible by 1. Nothing can be said about the first cipher.

Second cipher
A number is divisible by 2 if it is even, so if it ends on a 0, 2, 4, 6, or 8. The 0 has already been used, so the second cipher of the requested number is a 2, 4, 6, or 8.

The fourth, sixth, and eighth ciphers of the requested number must also be divisible by two, so these ciphers must be 2, 4, 6, or 8 too. The ciphers on the first, third, fifth, seventh, and ninth positions of the requested number can only be 1, 3, 5, 7, or 9.

Third cipher
A number is divisible by 3 if the sum of its ciphers is divisible by 3. Below all possibilities for the first thee ciphers of the requested number (first and third ciphers are 1, 3, 5, 7, or 9, second cipher is 2, 4, 6, or 8, and the sum of the ciphers is divisible by 3):

123 723 147 183 783
129 729 741 189 789
321 921 369 381 981
327 927 963 387 987

Fourth cipher
A number is divisible by 4 if:

* the number ends on a 0, 4, or 8 and the last but one cipher is even, or
* the number ends on a 2 or 6 and the last but one cipher is odd.

The third cipher of the requested number is odd, so the fourth cipher can only be a 2 or 6. Below are all possibilities for the first four ciphers of the requested number:

1236 9216 3692 3812 7892
1296 9276 9632 3816 7896
3216 1472 1832 3872 9812
3276 1476 1836 3876 9816
7236 7412 1892 7832 9872
7296 7416 1896 7836 9876

Sixth cipher
A number is divisible by 6 if it is divisible by 2 and 3, so if it ends on a 0, 2, 4, 6, or 8, and the sum of the ciphers is divisible by 3. The first three ciphers of the requested number are already divisible by 3, so the sum of the fourth, fifth, and sixth ciphers must be divisible by 3 too. Below are the two possibilities for the fourth, fifth, and sixth ciphers of the requested number (fourth cipher is 2, or 6, fifth cipher is 5, sixth cipher is 2, 4, 6, or 8, and the sum of the ciphers is divisible by 3):

258 654

Combined with what we already know about the first five ciphers, this gives the following possibilities for the first sixth ciphers of the requested number:

123654 723654 147258 183654 783654
129654 729654 741258 189654 789654
321654 921654 369258 381654 981654
327654 927654 963258 387654 987654

Eighth cipher
A number is divisible by 8 if:

* the number formed by the last two ciphers is divisible by 8 and the last but two cipher is even, or
* the number formed by the last two ciphers minus 4 is divisible by 8 and the last but two cipher is odd.

The last but two cipher is the sixth cipher of the requested number, and is a 4 or 8. Therefore, the number formed by the seventh and eighth ciphers must be divisible by 8. In addition, we know that the seventh cipher must be odd. These are the possible combinations:

16 32 56 72 96

Combined with what we already know about the first six ciphers, this gives the following possibilities for the first eight ciphers of the requested number:

18365472 78965432
18965432 98165432
18965472 98165472
38165472 98765432

Seventh cipher
The number formed by the first seven ciphers of the requested number must be divisible by 7. For the numbers shown above, this only holds for the number 38165472.

Ninth cipher
For the ninth cipher, only the cipher 9 remains. Note that every number formed by the ciphers 1 up to 9 is divisible by 9. A number is divisible by 9 if the sum of its ciphers is divisible by 9. The sum of the ciphers 1 up to 9 is 45, which is divisible by 9.

Conclusion
The requested number is 3816547290.

second Answer: The number is 69: 692 = 4761 and 693 = 328509.


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