Number Inception

The whole puzzle is based on the process of elimination. The first things we can deduce are: - Any number divisible by 10 ends in a 0, hence xxxxxxxxx0 - Any number divisible by 5 ends in a 0 or a 5. We have already used the 0, hence xxxx5xxxx0 - The second, fourth, sixth and eight digit can be 2, 4, 6 or 8 - The third, fifth, seventh and ninth digit can be 1, 3, 7, 9 (5 is already used) - The first number can be any

Now the real fun begins

Any number divisible by 3 has digits that, when added up, also divide by 3. (e.g. 171 -> 1+7+1=9=TRUE). Therefor the possibilities are as follows, 123 129 147 183 189 321 327 369 381 387 723 729 741 783 789 921 927 963 981 987 (Notice they all end in 1, 3 or 7 and follow a 2, 4 or 8)

Any number divisible by 4 either, - Ends in a 0, 4, or 8 after a even digit (e.g. 24) - Ends in a 2 or 6 after an odd digit (e.g. 36) As the 'three digit' number before is odd (we worked out it is 1, 3 or 7), the fourth value can only be a 2 or a 6. This still leaves us with a lot combinations.

A number divisible by 6 ends in 0, 2, 4, 6 or 8, but we have already used the 0. We already know the fifth digit has to be a five and the first 3 digits add up to be a multiple of 3. Likewise, any number divisible by 6 also has a sum of digits that can be divided by 3 (because 6 is divisible by 3 and 2). As the first three digits have to add up to a multiple of 3, logic dictates that the 'fourth, fifth and sixth values' will have to do the same. Due to the fifth digit being a 5, there are only two options: 258 or 654 (note: 852 and 456 don't work) So our number can be either xxx258xxx0 or xxx654xxx0, which helps us narrow down our overall possibilities:

123654xxx0 129654xxx0 147258xxx0 183654xxx0 189654xxx0 321654xxx0 327654xxx0 369258xxx0 381654xxx0 387654xxx0 723654xxx0 729654xxx0 741258xxx0 783654xxx0 789654xxx0 921654xxx0 927654xxx0 963258xxx0 981654xxx0 987654xxx0

For numbers divisible by 8, - the value formed by the last two digits is divisible by 8 and the digit before is even (e.g. 264) or... - the value formed by the last two digits minus 4 is divisible by 8 and the digit before is odd (e.g. 104) We already know the digit before (aka the sixth number) is either a 4 or an 8, so it must be even. Therefor the last two digits must be exactly divisible by 8. Moreover we know the seventh digit must be odd and through looking what numbers are used elsewhere (e.g. 0, 2&8, etc.) we can narrow down our options to: 16, 32, 72, 96

This leaves us with ten possibilities, having simply infered the rest of the numbers from our various rules:

14725896x0 18365472x0 18965432x0 18965472x0 38165472x0 74125896x0 78965432x0 98165432x0 98165472x0 98765432x0

The seventh digit has to be divisible by 7. There aren't any tricks here, we need to check all of them. The only one that fits is 38165472x0

If we where lazy we could just fill in the missing number, but let's prove that it works. Any number divisible by 9 has digits which produce the sum of a number that is also divisible by 9 (e.g. 9126 -> 9+1+2+6) Hence we can check: 3+8+1+6+5+4+7+2= 36 36 + 9 (the missing number) = 45 = TRUE

Now we can safely fill it in and get: 3816547290