The rule
digital root = keep adding the digits until one is left
Why it works
Adding the digits, and adding again until a single digit remains, gives the digital root. It equals the number's remainder modulo 9, with 9 standing in for 0, which is why it detects divisibility by 3 and 9 so directly.
How to do it by hand
- Add all the digits together
- If the result has more than one digit, add again
- Repeat until a single digit is left
- A root of 3, 6 or 9 means divisible by 3; a root of 9 means divisible by 9
What is worth knowing
The link to 9 is not a coincidence: 10 leaves remainder 1 when divided by 9, so 100, 1000 and every power of ten do too. That means a number and its digit sum always have the same remainder modulo 9, which is exactly the divisibility test. The same fact powers casting out nines, a checking method used by merchants for centuries before calculators: if the digital roots of your operands do not match the digital root of your answer, the arithmetic is wrong. It catches most errors, though not digit transpositions, which leave the sum unchanged.