The number formed by writing the digit d three times, ddd (like 111, 222, 333), is the same as d × 111.
For how many single digits d (from 1 to 9) is ddd divisible by 9?

The digit sum of ddd is d + d + d = 3d. For divisibility by 9, 3d must be a multiple of 9, which means d must be a multiple of 3. From 1 to 9 the multiples of 3 are d = 3, 6, 9 — that is 3 digits (333, 666, 999 are each divisible by 9).