Reasoning with integer numbers.
We must find the largest possible integer number with only even digits that is:
* less than 1000
* multiple of 9
First, let's easily identify when a number is a multiple of 9.
A simple test to determine if a number is multiple of 9 is to sum its digits. If the sum is a multiple of 9, then the original number was also a multiple of 9.
For example, 456 is not a multiple of 9 because 4+5+6=15 and 15 is not
But the number 459 is a multiple of 9 because 4+5+9=18 and 18 is a multiple of 9.
Now since we need our number to be as large as possible closer to 1000 and made of only even numbers, our first possible choice would be 8888.
But 8+8+8+8=32 and it's not a multiple of 9.
If we fix the first two 8's and move around the last two, we have the following:
8+8 = 16
The next multiples of 9 from 16 on are 18, 27, and 36.
If we use 36, we would need the last two digits to sum 36-16=20 and it's not possible.
If we use 27, the last two digits would not be even.
We must use 18, and the last two digits must add up to 18-16=2
The largest possible number to sum 2 and made only of even numbers is 20, thus our answer is:
8820