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.
This is not hard by hand, starting at the left side
In order for DI to be divisible by 8, DI must be
16,
24,
32,
48,
56,
64 or
72
In order for IS to be divisible by 7, DIS must be
163,
321,
328,
563,
642 or
721
In order for SC to be divisible by 6, DISC must be
3218,
3284 or
7218
In order for CO to be divisible by 5, O must be 5
In order for OV to be divisible by 4, DISCOV must be
321856,
328456 or
721856
In order for VE to be divisible by 3, DISCOVE must be
7218563
So the final and only answer is
72185634
Manual Solution (spoiler)
