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)

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