To find the remainder when divided by 9, you can cast out 999's to find the remainder when divided by 999. The reason this works is the same: 1000 = 1 modulo 999, so xyz000...000 = xyz modulo 999. But instead of just adding up the digits, you have to add three-digit groups.
Take for instance the number 12345678. Break this into three-digit groups (starting from the right):
12 345 678
Take the sum:
12 + 345 + 678 = 1035
Reduce modulo 999, using the same trick:
1035 = 1 + 035 = 36 mod 999
So 12345678 = 36 mod 999.
In your case, we have
678 967 896 789
repeated 25 times:
25 * (678 + 967 + 896 + 789) = 25 * 3330 = 83250
So 67896789...6789 = 83250 = 83 + 250 = 333 modulo 999.
