A test for divisibility by 11 is to add alternate
digits:

1 + 3 + * + 7 = 11 + *; 2 + 4 + 6 + 8 = 20.

If the original number is a multiple of 11 then these two
totals will be the same or will differ by a multiple of
11. In this case, 11 + * = 20 gives * = 9.

Or, you can solve it without knowing a rule as follows:

1234*678 = 12340678 + 1000* = (11 x 1121879 +9) + 11 x 90* +
10*

and hence is divisible by 11 if and only if 10* + 9 is
divisible by 11. So * = 9.

*This problem is taken from the UKMT Mathematical Challenges.*