Digital roots

Take a number and add its digits. If the answer has more than one digit, add again. Keep going until one digit is left. That digit is the digital root.

For 489710: 4 + 8 + 9 + 7 + 1 + 0 = 29. Then 2 + 9 = 11. Then 1 + 1 = 2. So the digital root of 489710 is 2.

You've met this already: it's exactly what we did to find the remainder on division by 9. That tells you what a digital root really is.

Since every place value is one more than a multiple of 9, the digit sum keeps the same remainder on division by 9 as the number itself. So the digital root is that remainder, with one quirk: a multiple of 9 comes out as 9 rather than 0.

That gives an instant test. The digital root of every multiple of 9 is 9. The digital root tells you the remainder on division by 9, and, since 3 divides 9, about division by 3 too (a digital root of 3, 6 or 9 means divisible by 3). And multiples of 9 inside a sum simply drop out: in 9a + 36b + 13, only the 13 matters.

Write the digital roots of twelve consecutive numbers and you'll see them cycle 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3: a loop of nine, just as remainders should.

Then try consecutive multiples of 3, of 4, of 6. Each gives its own repeating pattern, and each pattern has a reason. The book asks you to make conjectures: look for a link between a number's parity and its digital root, and between its digital root and its remainder on division by 3 or 9.

This isn't a modern party trick. Aryabhata II, in the Mahāsiddhānta of about 950 CE, describes finding the digital root by adding the digits again and again, and it was used to check arithmetic. If the digital roots don't match after a calculation, there's a mistake somewhere.