We have a number AAAA...AA, where all the digits are the same, and we need the number to be divisible by 1, 2, 3, 4, 6, 7, 8 and 9. If the number is divisible by 8, it will definitely be divisible by 4 and by 2. If it is divisible by 8 and 9, it will definitely be divisible by 6 and by 3. So really all we need is for our number to be divisible by 7, 8 and 9.
Now, for a number to be divisible by 8, its last three digits must make a number divisible by 8. Clearly our digit A must then be even. But the numbers 222, 444 and 666 are not multiples of 8. So the only possible value of A is 8, and we know our number looks like 8888....888.
For this number to be a multiple of 9, its digits will need to sum to a multiple of 9. That will only happen if we have 9 or 18 or 27 etc identical digits in our number. So the answer must be C or E.
To decide if C is correct, we just need to know if 888,888,888 is divisible by 7. There is some divisibility test for 7 that is completely useless for GMAT purposes (and in most areas of math in general - I have a Masters specifically in Number Theory and have literally never used that test once in my life). I imagine it would be helpful for this precise question, because if I recall correctly, it involves subtracting even and odd placed digits, which would be easy to do for this kind of number, so that divisibility test probably produces an answer instantly here. But because I don't know that test and have no reason to learn it, I'd instead just subtract large multiples of 7 from 888,888,888 until I got to a familiar number -- if we're starting from a multiple of 7, we'll need to get new smaller multiples of 7 every time we subtract a multiple of 7. So starting from 888,888,888, I subtracted 777,777,777 to get 111,111,111, then subtracted 105,000,000 to get 6,111,111 then subtracted 5,600,000 to get 511,111, then subtracted 490,000 to get 21,111, then subtracted 21,000 to get 111. That's not a multiple of 7, so 888,888,888 cannot be either. That leaves E as the only possible answer.