A Will to be Interpreted
Molly McKee
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A nine-digit number is formed using each of the digits 1,2,3,...,9 exactly once. For n = 1,2,3,...,9, n divides the first n digits of the number.
The Problem
When n = 1, n must divide the first digit, therefore the first digit can only be 1
  1. 1 - - - - - - - -
 
The second digit can be 2, 4, 6, or 8, since all of these numbers are divisible by 2
  1. Let’s use 2 for right now
  2. 1 2 - - - - - - -
 
Recall casting out 3’s,
If x is the third digit, then 1 + 2 + x is a multiple of 3
Using this rule, only 3, 6, or 9 can be used
  1. Let’s use 3 for right now
  2. 1 2 3 - - - - - -
 
Of the numbers remaining to be used, the fourth digit can only be 6
  1. 1 2 3 6 - - - - -
 
Similar to n=1, when n=5, the fifth digit must be either 5 or 0; zero is not an option
  1. 1 2 3 6 5 - - - -
 
There are only four numbers left, 4, 7, 8 and 9
  1. Let’s use 4 for right now
  2. 1 2 3 6 5 4 - - -
 
There are only 3 numbers left, 7, 8 and 9, but none of then seem to work
  1. Is there another option for the sixth digit? No.
  2. The fifth digit cannot be changed
  3. The fourth digit was dependent on the choice of digits two and three
 
We’ll begin again
  1. 1 - - - 5 - - - -, what we know so far
  2. 1 4 - - 5 - - - -, this leave no value for the third digit
  3. 1 6 - - 5 - - - - , the only possible value for the third digit is 5, which cannot work
  4. The second digit must be 8
  5. 1 8 3 - 5 - - - -, the third digit must be 3
  6. 1 8 3 4 5 - - - -
  7. 1 8 3 4 5 6 - - -
  8. 1 8 3 4 5 6 7 - -
  9. 1 8 3 4 5 6 7 2 -
  10. 1 8 3 4 5 6 7 2 9
 
Notice that no matter what the order of the digits are, the sum is always 45.
This means that a nine digit number using the numbers 1 through 9 will always be divisible by 9.
Also notice, the numbers are in sequential order, with the exception of 2 and 8.