Solution to:
Notable Number
We construct the number digit by digit.
Tenth digit
A number is divisible by 10 if it ends on a 0.
Therefore, the tenth digit of the requested number must be a 0.
Fifth digit
A number is divisible by 5 if it ends on a 0 or 5.
The 0 has already been used, so the fifth digit of the requested number is a 5.
First digit
A number is always divisible by 1.
Nothing can be said about the first digit.
Second digit
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 digit of the requested number is a 2, 4, 6, or 8.
The fourth, sixth, and eighth digits of the requested number must also be divisible by two, so these digits must be 2, 4, 6, or 8 too.
The digits on the first, third, fifth, seventh, and ninth positions of the requested number can only be 1, 3, 5, 7, or 9.
Third digit
A number is divisible by 3 if the sum of its digits is divisible by 3.
Below all possibilities for the first three digits of the requested number (first and third digits are 1, 3, 5, 7, or 9, second digit is 2, 4, 6, or 8, and the sum of the digits 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 digit
A number is divisible by 4 if:
 the number ends on a 0, 4, or 8 and the lastbutone digit is even, or
 the number ends on a 2 or 6 and the lastbutone digit is odd.
The third digit of the requested number is odd, so the fourth digit can only be a 2 or 6.
Below are all possibilities for the first four digits 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 digit
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 digits is divisible by 3.
The first three digits of the requested number are already divisible by 3, so the sum of the fourth, fifth, and sixth digits must be divisible by 3 too.
Below are the two possibilities for the fourth, fifth, and sixth digits of the requested number (fourth digit is 2, or 6, fifth digit is 5, sixth digit is 2, 4, 6, or 8, and the sum of the digits is divisible by 3):
258 654
Combined with what we already know about the first five digits, this gives the following possibilities for the first sixth digits 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 digit
A number is divisible by 8 if:
 the number formed by the last two digits is divisible by 8 and the lastbuttwo digit is even, or
 the number formed by the last two digits minus 4 is divisible by 8 and the lastbuttwo digit is odd.
The lastbuttwo digit is the sixth digit of the requested number, and is a 4 or 8.
Therefore, the number formed by the seventh and eighth digits must be divisible by 8.
In addition, we know that the seventh digit must be odd.
These are the possible combinations:
16 32 56 72 96
Combined with what we already know about the first six digits, this gives the following possibilities for the first eight digits of the requested number:
18365472 74125896
18965432 78965432
18965472 98165432
38165472 98165472
14725896 98765432
Seventh digit
The number formed by the first seven digits of the requested number must be divisible by 7.
For the numbers shown above, this only holds for the number 38165472.
Ninth digit
For the ninth digit, only the digit 9 remains.
Note that every number formed by the digits 1 up to 9 is divisible by 9.
A number is divisible by 9 if the sum of its digits is divisible by 9.
The sum of the digits 1 up to 9 is 45, which is divisible by 9.
Conclusion
The requested number is 3816547290.
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