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See if you are able to find a nine-digit number that meets the following conditions:

• All digits 1 through 9 must appear only once
• The number must be divisible by 9
• If we delete the last digit on the right, the number must be divisible by 8
• If we delete the last two digits on the right, the number must be divisible by 7
• If we delete the last three digits on the right, the number must be divisible by 6
• If we delete the last four digits on the right, the number must be divisible by 5
• If we delete the last five digits on the right, the number must be divisible by 4
• If we remove the last six digits from the right, the number must be divisible by 3
• If we delete the last seven digits on the right, the number must be divisible by 2
• If we delete the last eight digits on the right, the number must be divisible by 1

#### Solution

Let's call the ABCDEFGHI number where each letter will represent a different digit. It is clear that the digits B, D, F and H must be even since they correspond to the last digit of the numbers that must be divisible by even numbers (2, 4, 6 and 8). The rest will therefore be odd digits since we know that you must include all the numbers from 1 to 9.

Since ABCDE is divisible by 5, we know that E has to be equal to 5.

Since ABCD is divisible by 4, it will also be fulfilled that CD will be divisible by 4 and GH will be divisible by 8 (since FGH will be divisible by 8 and F is even).

Because C and G are odd, D and H must be 2 and 6 but not necessarily, in this order.

We know that ABC is divisible by 3, that ABCDEF is divisible by 6 and therefore also by 3 and that ABCDEFGHI is divisible by 9 and therefore also by 3 so that A + B + C, D + E + will be fulfilled F and G + H + I are divisible by three.

If we assume for example D = 2, then it would be true that F = 8, H = 6 and B = 4. A + 4 + C is divisible by 3, therefore, A and C must be 1 and 7 or vice versa and G and I must be 3 and 9 or vice versa. GH is divisible by 8, therefore it must be agreed that G = 9 and from the previous conclusion we obtain that I = 3. In this case the possible numbers 1472589 and 7412589 are not divisible by 7. Therefore it must be met what D = 6 Where do we deduce that F = 4, H = 2, B = 8.

G + 2 is divisible by 8, therefore, G can only be 7 or 3.

A + 8 + C is divisible by 3 and therefore the values ​​of A and C must be one 1 or 7 and the other 3 or 9.

If we assume for example G = 3, then A or C must be 9 and the other must be 1 or 7. But none of the numbers 1896543, 7896543, 9816543 and 9876543 are divisible by 7. Therefore G = 7 and then A or C must be equal to 1 and the other 3 or 9. Of the possible numbers 1836547, 1896547, 3816547 and 9816547, 3816547 only the latter is divisible by 7 (the quotient is 545221). Thus, The number we are looking for is 381654729.

1. Taggart

2. Akinojora

There is something in this. Thank you very much for your help with this issue. I did not know it.

3. Mibei

I congratulate, a brilliant idea and it is duly