What's the first number greater than one million that's divisible by 9?

Finding the first number after one million that is a multiple of nine is simpler than it sounds, thanks to a clever mathematical shortcut. The divisibility rule for nine states that if the sum of a number's digits is divisible by 9, then the number itself is also divisible by 9. To find our target, we just need to check the numbers in order starting from 1,000,001. The sum of the digits for 1,000,001 is 2, for 1,000,002 it's 3, and so on. We don't find a sum divisible by 9 until we reach 1,000,008. The sum of its digits (1 + 0 + 0 + 0 + 0 + 0 + 8) is 9, which is perfectly divisible by 9, making 1,000,008 our answer.

This handy trick, sometimes known as "casting out nines," works because of our base-10 number system. Every power of ten (10, 100, 1000, etc.) is just one more than a multiple of nine (e.g., 100 = 99 + 1). When you break a number down by its place values, the parts that are multiples of 9 can be effectively ignored when checking for divisibility, leaving you with just the sum of the face-value digits. If that leftover sum is a multiple of 9, the whole original number must have been, too. A very similar rule applies for divisibility by 3, where the sum of the digits must be divisible by 3.