The digital root is the basis of “casting out nines”, an old way of checking sums and products. A number is divisible by 9 exactly when its digital root is 9, and by 3 when its digital root is 3, 6 or 9.
How to use the Digital Root Calculator
- Number
Results update as you type, and everything is calculated in your browser.
How it works
Instead of adding digits again and again, the calculator uses the fact that a number and the sum of its digits leave the same remainder when divided by 9.
Example values
How the main result changes when Number changes and everything else stays at the example values above:
| Number | Digital root |
|---|---|
| 32,768 | 8 |
| 49,152 | 3 |
| 65,536 | 7 |
| 81,920 | 2 |
| 98,304 | 6 |
| 131,072 | 5 |
Formula
dr(n) = 0 if n = 0, otherwise 1 + (n − 1) mod 9
Worked example
Inputs — Number: 65,536.
Results — Digital root = 7; Remainder mod 9 = 7; Divisible by 9? = No; Divisible by 3? = No.
Key terms
- Digital root
- The single digit obtained by repeatedly summing digits.
Frequently asked questions
What is the digital root of 65,536?
6 + 5 + 5 + 3 + 6 = 25, then 2 + 5 = 7.
What is casting out nines?
Checking arithmetic by comparing the digital roots of the operands and the result.
What is additive persistence?
The number of digit-summing steps needed to reach one digit.
What is the digital root of a multiple of 9?
Always 9 (except 0).