Determine all possible value(s) of an eight-digit base ten positive integer having the form DISCOVER that uses each of the nonzero digits from 1 to 8 exactly once and satisfies all of these conditions:

• ER is divisible by 2
• VE is divisible by 3
• OV is divisible by 4
• CO is divisible by 5
• SC is divisible by 6
• IS is divisible by 7
• DI is divisible by 8

Note: Think of this problem as a reverse of Ten-Digit Numbers.

(In reply to Manual Solution (spoiler) by Steve Herman)

I won't add to what you have already deduced as I had followed an identical train of reasoning but hadn't documented it.

There was one thing that I did not immediately notice, and is quite apparent is that  DISCOVER is written vertically up the left hand column until the final E and then the R is shifted to the right.

Beginning again at the bottom of the left hand column with the D, the rest of the word DISCOVER is written upwards in the right hand column.

This means that D and R only occur once while the others are represented twice.  In the analysis the Units digit of each lower number in the column pair becomes the Tens digit in the number directly above.

Steve probably didn't consider an explanation as the above but he certainly took it into account.

Nice documentation Steve.

Posted by brianjn on 2009-06-29 10:16:18