Digital root

The idea

The single-digit number obtained by adding the digits of a number repeatedly is its digital root; the digital root of every multiple of 9 is 9.

Digits down to one digit

Take 489710. Add its digits: 4 + 8 + 9 + 7 + 1 + 0 = 29. Add again: 2 + 9 = 11. Add again: 1 + 1 = 2. The single digit 2 is the digital root of 489710.

What digital root do you expect for the multiples of 9: 9, 18, 27, 36 …?

• 9 every time
• 0 every time
• Different each time

Colour by digital root

The numbers are in rows of nine, coloured by their digital root. Tap a number to follow its digits down to one digit. A dashed outline shows each step of the chain.

Multiples of 9 end at 9

The single-digit number obtained by adding the digits of a number repeatedly is its digital root; the digital root of every multiple of 9 is 9.

Why 9 and not 0? A multiple of 9 leaves remainder 0 when divided by 9. But the digits of a positive number add up to a positive number, so the single digit left is never 0. It must be 9.

In the grid, every step of a chain stays in its column: adding the digits only takes away multiples of 9. The multiples of 9 fill the last column.

Aryabhata II (about 950 CE) wrote about adding digits repeatedly in his Mahāsiddhānta. The method was used to check calculations.

Notes

The single-digit number obtained by adding the digits of a number repeatedly is its digital root; the digital root of every multiple of 9 is 9.

Check yourself

What is the digital root of 68975?

Answer: 8

6 + 8 + 9 + 7 + 5 = 35 and 3 + 5 = 8. So the digital root of 68975 is 8.

Which of these numbers has digital root 9?

Why can the digital root of a multiple of 9 never be 0?

How many numbers from 1 to 50 have digital root 9?

Answer: 5

The multiples of 9 up to 50 are 9, 18, 27, 36 and 45. That is 5 numbers.

• 9001. It starts with 9, but 9 + 0 + 0 + 1 = 10 and 1 + 0 = 1. Its digital root is 1.
• 1809 — correct. Yes! 1 + 8 + 0 + 9 = 18 and 1 + 8 = 9. So 1809 is a multiple of 9 (1809 = 9 × 201).
• 2900. 2 + 9 + 0 + 0 = 11 and 1 + 1 = 2. Its digital root is 2.
• 4096. 4 + 0 + 9 + 6 = 19, 1 + 9 = 10, 1 + 0 = 1. Its digital root is 1.
• Because 0 is not a digit. 0 is a digit. The reason is about adding digits of a positive number.
• Because the digits of a positive number add up to a positive number, so the single digit is at least 1 — correct. Yes! It is a multiple of 9, so it is not 1 to 8. The only single digit left is 9.
• Because multiples of 9 are not divisible by 9. Multiples of 9 are divisible by 9, with remainder 0.