What is the largest positive integer with only even digits that is less than 10,000 and is a multiple of 9?
There are 24=16 four-digit numbers containing only the digits 1 and 2. The same for all other of the (92)=9⋅82=36 combinations of 2 of the 9 possible digits. If you add all those together, you will have counted the number "1111" and "2222" and so on too often. The highest possible integer in this context is 8,820.
Here's another (less neat ) solution
We can guess that the answer might be 88 _ _
Note that 8901 is a multiple of 9 because the sum of its digits is divisible by 9
We need to subtract some odd multiple of 9 from this to produce an integer with all even digits
The possibilities are 9, 27, 45, 63, 81
81 is the one we need
8901 - 81 =
8820 !!! (as the guest found)