Every three-digit number whose digits are all the same — 111, 222, 333 and so on — has a common divisor. What is the LARGEST number that divides all nine of them?

A number with all three digits equal to a is $100a+10a+a=111a$, so each one is 111 times a single digit. Since 111 is itself on the list, no larger number can divide them all. The distractors 3 and 37 are the prime factors of $111=3\times 37$: each really does divide every such number, but neither is the largest. And 11 divides none of them, because $111a$ leaves remainder a on division by 11, and a runs only from 1 to 9.