without a calculator, divide 12,345,678 by 9
now divide 123,456,789 by 9 (why was this easy?)
use the digits 1 to 9 once only
in the sum:
students who have some familiarity with divisibility tests (or have just done work on this) will appreciate that the single digit cannot be 3, 6 or 9 because the result isn't divisible by 3
or 5 because the result doesn't end with 5 or 0
or 1, which works but this is a trivial case
all other options for the single digit could work...
an intention is that students divide 98765432 by each of the remaining four digits to see if they work...
2 nearly does
4 nearly does
7 doesn't work without a remainder
and 8 does work
so there is only one solution (discounting the trivial case)