# Digital Root Calculator

URL: https://speedycalculator.com/calculators/math/digital-root/
Last updated: 2026-10-30

> The digital root calculator repeatedly adds the digits of a number until one digit is left – the digital root – and gives the shortcut formula 1 + (n − 1) mod 9.

## Inputs

- Number

## Results

- Digital root
- Remainder mod 9
- Divisible by 9?
- Divisible by 3?

## Formula

```
dr(n) = 0 if n = 0, otherwise 1 + (n − 1) mod 9
```

## Explanation

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:

NumberDigital root32,768849,152365,536781,920298,3046131,0725

## 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.

## FAQ

**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).


## Sources

- [OpenStax – Prealgebra 2e](https://openstax.org/details/books/prealgebra-2e)
- [Wolfram MathWorld – Digital root](https://mathworld.wolfram.com/DigitalRoot.html)